}}0^ 


IN  MEMORIAM 
FLORIAN  CAJORl 


Digitized  by  the  Internet  Archive 

in  2008  with  funding  from 

IVIicrosoft  Corporation 


http://www.archive.org/details/advancedalgebraOOschurich 


ADVANCED  ALGEBRA 


^2^^^ 


ADVANCED  ALGEBRA 


BY 


ARTHUR   SCHULTZE,   Ph,D. 

ASSISTANT   PROFESSOR  OF  MATHEMATICS,   NEW  YORK  UNIVERSITY 

HEAD  OF  THE   MATHEMATICAL  DEPARTMENT,   HIGH 

SCHOOL  OF  COMMERCE,   NEW  YORK  CITY 


THE   MACMILLAN   COMPANY 

LONDON:  MACMILLAN  &  CO.,  Ltd. 

1906 

All  rights  reserved 


Copyright,  1905,  1906, 
By  the  MACMILLAN  COMPANY. 


Set  up  and  electrotyped.     Published  January,  1906. 
Reprinted  September,  twice,  1906. 


PREFACE 

Most  teachers  of  mathematics  agree  that  a  number  of  the 
topics  ordinarily  taught  to  classes  in  advanced  algebra  may  be 
omitted  without  injury  to  the  course.  Some  of  these  topics, 
such  as  multiple  roots,  Sturm's  theorem,  etc.,  can  be  more  sat- 
isfactorily taken  up  after  the  student  is  familiar  with  calculus, 
while  others,  such  as  recurring  series,  continued  fractions,  etc., 
are  so  seldom  applied  in  higher  mathematics  that  they  may  be 
entirely  omitted. 

In  accordance  with  this  view,  the  College  Entrance  Exami- 
nation Board  has  considerably  reduced  the  number  of  topics 
required  in  advanced  algebra.  All  subjects  no  longer  required 
for  the  examinations  of  this  Board  are  omitted  from  the  regu- 
lar course  of  this  book,  with  the  exception  of  inequalities, 
which  is  retained,  since  familiarity  with  the  symbols  of  in- 
equaUty  seems  to  be  necessary  for  future  work.  If,  however, 
a  subject  appears  too  important  for  entire  omission,  it  is  placed 
in  the  Appendix.  This  is  done  in  the  case  of  indeterminate 
equations,  logarithms,  summation  of  series,  and  some  other 
subjects. 

On  the  other  hand,  graphical  methods  are  emphasized  more 
than  is  usual  in  text-books  of  this  grade.  The  graphical 
method  for  solving  cubics  given  in  Section  578  is  not  met  with 
in  any  other  text-book,  and  the  method  for  representing  a 
cubic  function  by  means  of  one  standard  curve  (Section  583) 
is  entirely  new.  Summation  of  series  is  also  treated  in  a 
novel  manner  (Appendix  IX).  While  the  method  given  is 
almost  identical  with  that  used  in  many  text-books,  it  is  here 
presented  in  a  more  practical  form,  which  makes  it  applicable 
to  all  cases. 

V 


i\^80(>l43 


VI  PREFACE 

The  first  twenty-two  chapters  are  identical  with  the  author's 
"  Elementary  Algebra/'  whose  general  plan  and  scope  are  stated 
in  its  preface  as  follows : 

"  The  author  has  aimed  to  make  this  treatment  of  elemen- 
tary algebra  simple  and  practical,  without,  however,  sacrificing 
scientific  accuracy  and  thoroughness. 

"  Particular  care  has  been  bestowed  upon  those  chapters  which 
in  the  customary  courses  offer  the  greatest  difficulties  to  the 
beginner,  especially  problems  and  factoring.  The  presentation 
of  problems  as  given  in  Chapter  V  will  be  found  to  be  quite 
a  departure  from  the  customary  way  of  treating  the  subject, 
and  it  is  hoped  that  this  treatment  will  materially  diminish 
the  difficulty  of  this  topic  for  young  students. 

"  In  factoring,  instead  of  the  usual  multiplicity  of  cases,  com- 
paratively few  methods  are  given,  but  these  few  are  treated 
thoroughly.  The  cross-product  method  for  factoring  quad- 
ratic trinomials  has  been  simplified  by  considering  the  common 
monomial  factors  (§  116,  4) ;  and  in  this  form  the  method 
seemed  to  be  preferable  to  the  other  prevailing  methods.  The 
criticism  that  the  cross-product  method  is  based  upon  guessing 
has  no  value,  since  all  other  devices  are  equally  based  upon 
guessing;  in  fact,  these  methods  have  to  be  empirical  until 
quadratic  equations  furnish  a  scientific  means  of  factoring. 

"  Applications  taken  from  geometry,  physics,  and  commercial 
life  are  numerous,  but  care  has  been  taken  not  to  introduce 
illustrations  so  complex  as  to  require  the  expenditure  of  time 
for  the  teaching  of  physics  or  geometry.  In  cases,  however, 
in  which  a  physical  or  geometric  formula  produced  an  example 
equally  good  as  the  putting  together  of  symbols  at  random, 
the  formula  has  been  used,  as  in  numerical  substitution,  pro- 
portion, literal  equations,  etc. 

"  The  book  is  designed  to  meet  the  requirements  for  admis- 
sion to  our  best  universities  and  colleges,  in  particular  the  re- 
quirements of  the  College  Entrance  Examination  Board.  This 
made  it  necessary  to  introduce  the  theory  of  proportions  and 


PREFACE  vii 

graphical  methods  into  the  first  year's  work,  an  innovation 
which  seems  to  mark  a  distinct  gain  from  the  pedagogical 
point  of  view. 

"  By  studying  proportions  during  the  first  year's  work,  the 
student  will  be  able  to  utilize  this  knowledge  where  it  is  most 
needed,  viz.  in  geometry  ;  while  in  the  usual  course  proportions 
are  studied  a  long  time  after  their  principal  application. 

"  Graphical  methods  have  not  only  a  great  practical  value, 
but  they  unquestionably  furnish  a  very  good  antidote  against 
*  the  tendency  of  school  algebra  to  degenerate  into  a  mechani- 
cal application  of  memorized  rules.'  This  topic  has  been 
represented  in  a  simple,  elementary  way,  and  it  is  hoped  that 
some  of  the  modes  of  representation  given  will  be  considered 
improvements  upon  the  prevailing  methods,  e.g.  the  finding  of 
roots  to  several  decimal  places  (§  305)  and  the  solution  of 
quadratic  equations  (§  330).  The  entire  work  in  graphical 
methods  has  been  so  arranged  that  teachers  who  wish  a 
shorter  course  may  omit  these  chapters." 

The  author  desires  to  acknowledge  his  indebtedness  to 
Messrs.  William  P.  Manguse  and  B.  A.  Heydrick  for  the  care- 
ful reading  of  the  proofs  and  for  many  valuable  suggestions. 

ARTHUR   SCHULTZE. 
New  York, 
October,  1905. 


CONTENTS 

CHAPTER  I 

PAOB 

Introduction 1 

Algebraic  Solution  of  Problems 1 

Negative  Numbers 4 

Numbers  represented  by  Letters 6 

Factors,  Powers,  and  Roots ,        .  8 

Algebraic  Expressions  and  Numerical  Substitutions  .        .        •  12 

CHAPTER   II 

Addition,  Subtraction,  and  Parentheses    .....  18 

Addition  of  Monomials        .      " 18 

Addition  of  Polynomials 22 

Subtraction ....25 

Review  Exercise  I 28 

Signs  of  Aggregation 30 

Exercises  in  Algebraic  Expression 33 

CHAPTER  III 

Multiplication o        .        ,  36 

Multiplication  of  Algebraic  Numbers 35 

Multiplication  of  Monomials 39 

Multiplication  of  a  Polynomial  by  a  Monomial  ....  40 

Multiplication  of  Polynomials     .......  42 

Special  Cases  in  Multiplication 45 

CHAPTER  IV 

Division ..53 

Division  of  Monomials 54 

Division  of  a  Polynomial  by  a  Monomial 55 

Division  of  a  Polynomial  by  a  Polynomial          ....  56 

Special  Cases  in  Division    ...••••.  61 

ix 


CONTENTS 


CHAPTER  y 

PAGE 

Linear  Equations  and  Problems  .        ,  ,  ^  .        .        .  64 

Solution  of  Linear  Equations      .,  <,  =  «,.  66 

Symbolical  Expressions       .         .        .  o  o  .         o        »  69 

Problems  leading  to  Simple  Equations  o  o  .        o        »  76 

•    Miscellaneous  Problems      .        ,        ,  .  .  ^        .        ,  j^4 

CHAPTER  VI 

.        .        .  96 

Polynomials,  All  of  whose  Terms  contain  a  Com- 
mon Factor 97 

Quadratic  Trinomials  of  the  Form  x^  -\-  px  -\-  q       .  98 

Quadratic  Trinomials  of  the  Form  px'^  +  qx  -{■  r     .  101 

The  Square  of  a  Binomial  x^  ±xy  +y^    .        .        .  104 

The  Difference  of  Two  Squares        .         .         =         .  105 
The  Sum  or  Difference  of  Two  Cubes      o      • .        .107 

Type  VII.     Grouping  Terms .  109 

Summary  of  Factoring «        o  112 

CHAPTER   VII 

Highest  Common  Factor  and  Lowest  Common  Multiple  .  ,114 
Highest  Common  Factor  .  .  .  „  .  .  .  .114 
Lowest  Common  Multiple  .        .        .        o        .        ."       .        ^     118 

CHAPTER  VIII 


Fa 

CTORING 

Type 

I. 

Type 

IL 

Type 

III. 

Type 

IV. 

Type 

V. 

Type 

VI. 

Fractions 


Reduction  of  Fractions 

Addition  and  Subtraction  of  Fractions 

Multiplication  of  Fractions 

Division  of  Fractions  .        .        .        o 

Complex  Fractions      .... 

Review  Exercise  II     . 


122 
122 

128 
136 
138 
141 
143 


CHAPTER  IX 

Fractional  and  Literal  Equations      „        o        o        o        o        .  147 

Fractional  Equations  .        .        ,        <.        o        o        <.        o        .  147 

Literal  Equations        .         .        .        o 162 

Identical  Equations 156 

Problems  leading  to  Fractional  and  Literal  Equations       «        .  158 


CONTENTS  XI 


CHAPTER  X 

PAGE 

Batio  and  Proportion    .      - o  o  167 

Ratio 0         .        o  b  o  167 

Proportion o        o  .  .  169 

Review  Exercise  III o        •  •  .  181 


CHAPTER  XI 

Simultaneous  Equations  of  the  First  Degree  ....  184 

Elimination  by  Addition  or  Subtraction 185 

Elimination  by  Substitution 188 

Elimination  by  Comparison 189 

Literal  Simultaneous  Equations 194 

Simultaneous  Equations  involving  More  than  Two  Unknown 

Quantities 196 

Problems  leading  to  Simultaneous  Equations     ....  201 

Interpretation  of  Negative  Results  and  the  Forms  of    -,   -,   —  209 


CHAPTER  XII 

Involution oooeo  212 

Involution  of  Monomials     .        .        ,        »        o        o        .  o  212 

Involution  of  Binomials       .        .        ,        <>        o        o        o  o  214 

Involution  of  Polynomials  ...        o        ...  .  216 

CHAPTER   XIII 

Evolution o        o        .  .  218 

Evolution  of  Monomials o  219 

Evolution  of  Polynomials  and  Arithmetical  Numbers         o  .  220 

Review  Exercise  IV o        .        .  »  227 


CHAPTER  XIV 

The  Theory  of  Exponents •  280 

Fractional  and  Negative  Exponents 230 

The  Laws  for  Negative  and  Fractional  Exponents     •       •        .  236 


XU  CONTENTS 

CHAPTER  XV 

Radicals »  . 

Transformation  of  Radicals         .        o  t 
Addition  and  Subtraction  of  Radicals 

Multiplication  of  Radicals  .        .        o  o 

Division  of  Radicals    .        .        .        .  = 
Involution  and  Evolution  of  Radicals 
Square  Roots  of  Quadratic  Surds 

Radical  Equations       .         .         .         .  = 
Extraneous  Roots  and  Equivalent  Equations 


PAGB 

242 

243 
248 
250 
252 
257 
258 
261 
264 


CHAPTER   XVI 

The  Factor  Theorem     .        .        .        .        o       o        o        =  -     .  267 

CHAPTER  XVII 

Graphic  Representation  of  Functions  and  Equations      .        .  272 
Representation  of  Functions  of  One  Variable     ....  272 
Graphic  Solution  of  Equations  involving  One  Unknown  Quantity  278 
Graphic  Solution  of  Equations  involving  Two  Unknown  Quan- 
tities             .        o        .  284 

CHAPTER   XVIII 

Quadratic  Equations  involving  One  Unknown  Quantity          .  289 

Pure  Quadratic  Equations o        .  289 

Complete  Quadratic  Equations   .        o        .        ..        .        =        .  292 

Problems  involving  Quadratics   .        .        o        »        o        »        .  301 

Equations  in  the  Quadratic  Form       ,        .        <.        o        ^        c  304 

Graphic  Solution  of  Quadratic  Equations  .        «        »        o        c  308 

CHAPTER  XIX 

Simultaneous  Quadratic  Equations      .        .        .        o        »        .  312 

I.     Equations  solved  by  finding  ;r +/ and  jf —/      .        .        .  313 

II.     One  Equation  Linear,  the  Other  Quadratic        «         o        .  315 

III.  Homogeneous  Equations    ,        .         ,        .        o        .         .  316    \ 

IV.  Special  Devices  .        o        e        ......  319     \ 

Problems     .        o        »        o        o        »        »        o        o        o        »  325 


r 


CONTENTS  '  XUl 


CHAPTER   XX 

PAOK 

Properties  of  Quadratic  Equations 327 

Character  of  the  Roots 327 

Relation  between  Roots  and  Coefficients 329 

Factoring  of  Quadratic  Expressions 331 


•^  CHAPTER  XXI 

Progressions 334 

Arithmetic  Progression 334 

Geometric  Progression 340 

Infinite  Geometric  Progression 343 


CHAPTER  XXII 

Binomial  Theorem 346 

Proof  by  Mathematical  Induction 346 

Binomial  Theorem  for  Integral  Exponents  .        .        .        .  348 

Review  Exercise  V 354 


CHAPTER   XXIII 

Inequalities 368 

CHAPTER   XXIV 

Variables  and  Limits      .        .  364 

CHAPTER  XXV 

Imaginary  and  Complex  Numbers 870 

4—        Algebraic  Treatment  of  Complex  Numbers         ....  370 

0^  Graphic  Representation  of  Complex  Numbers    ....  376 

CHAPTER   XXVI 

Permutations  and  Combinations 386 

4  Permutations 387 

^^'  Combinations 394 


>Vc' 


k 


XIV  CONTENTS 


CHAPTER   XXVII 

PAGE 

Determinants  . 401 

Determinants  of  the  Second  Order 401 

Determinants  of  the  Third  Order 404 

Determinants  of  the  Fourth  Order 409 

General  Properties  of  Determinants 413 

Solution  of  Linear  Equations  and  Elimination    ....  424 

CHAPTER   XXVIII 

Theory  of  Equations .         .  431 

Synthetic  Division 431 

Application  of  the  Factor  Tlieorem 434 

Number  of  Roots 437 

Relations  between  the  Roots  an<l  the  C(  ellicients       .         .         .  440 

Incommensurable  and  Imaginary  Roots 445 

Transformation  of  Equations 448 

Descartes'  Rule  of  Signs 452 

Location  of  Roots 466 

CHAPTER   XXIX 

Solution  of  Higher  Equations      ...        .         .        .         .         .  463 

Commensurable  Roots 463 

Incommensurable  Roots 470 

Graphic  Solution  of  Cubic  Equations 484 

General  Solution  of  Cubic  Equations 492 

Solution  of  Biquadratics      .         .         .        .        .         .        .         .  496 

Reciprocal  Equations 497 

APPENDIX 

I.     Multiplication  by  Detached  Coefficients         ....  501 

II.     Additional  Cases  in  Factoring 502 

III.  Highest  Common  Factor  and  Lowest  Common  Multiple        .  505 

IV.  Cube  Roots  of  Polynomials  and  Arithmetic  Numbers    .         .  509 
V.     Indeterminate  Equations  of  the  First  Degree         .         .         .512 

VI.     Variation 516 

VII.     Logarithms .         .         .519 

VIII.     Compound  Interest  and  Annuities 538 

IX.     Summation  of  Series      ........  542 


ELEMENTARY   ALGEBRA 

CHAPTER  I 
INTRODUCTION 

1.  Algebra  treats  of  numbers  as  does  arithmetic,  but  for 
reasons  which  will  appear  later,  numbers  are  frequently  de- 
noted by  letters ;  as,  a,  m,  x. 

2.  Known  numbers  are  usually  represented  by  the  first  letters 
of  the  alphabet;  as,  a,  b,  c.  '  ■" 

Unknown  numbers  are  usually  represented  by  the  last  letters- 
of  the  alphabet ;  as,  x,  y,  z. 

3.  The  signs  of  addition,  subtraction,  multiplication,  division, 
and  equality  have  the  same  meaning  in  algebra  as  they  have  in 
arithmetic. 

4.  A  problem  is  a  question  proposed  for  solution. 

5.  An  equation  is  a  statement  expressing  the  equality  of  two 
quantities ;  as,  7  a;  =  56. 

ALGEBRAIC   SOLUTION  OF  PROBLEMS 

6.  Problem.  The  sum  of  two  numbers  is  56,  and  the  greater 
is  six  times  the  smaller.     Find  the  numbers. 

Let  X  =  the  smaller  number. 

Then  6x  =  the  greater  number, 

and  7x  =  the  sum  of  the  two  numbers. 

Therefore,         7  a;  =  56, 

x  =  Sj  the  smaller  number, 
and  6  a;  =  48,  the  greater  number. 

B  1 


2  ELEMENTARY  ALGEBRA 

7.  In  algebra,  problems  are  frequently  solved  by  denoting 
numbers  by  letters  and  by  expressing  the  problem  in  the  form 
of  an  equation. 

EXERCISE  1 

Solve  algebraically  the  following  problems : 

1.  A  man  sold  a  horse  and  carriage  for  $450,  receiving 
twice  as  much  for  the  horse  as  for  the  carriage.  How  much 
did  he  receive  for  the  carriage  ? 

2.  A,  B,  C,  and  D  buy  $  1050  worth  of  goods.  How  much 
does  A  take,  if  B  buys  twice  as  much  as  A,  C  three  times  as 
much  as  B,  and  J)  six  times  as  much  as  B  ? 

3.  Divide  $160  among  A,  B,  and  C  so  that  A  may  receive 
three  times  as  much  as  B,  and  C  three  times  as  much  as  A  and 
B  together. 

4.  The  sum  of  the  three  angles  of  any  triangle  is  180°.  If 
2  angles  of  a  triangle  are  equal,  and  the  remaining  angle  is 
twice  their  sum,  how  many  degrees  are  there  in  each  ? 

5.  A  and  B  own  a  farm  worth  $12,600,  and  A  has  invested 
half  as  much  capital  as  B.     How  much  has  A  invested  ? 

6.  A  pole  50  feet  high  was  broken  so  that  the  part  broken 
off  was  4  times  the  length  of  the  part  left  standing.  Find  the 
length  of  the  two  parts. 

7.  Four  men,  A,  B,  C,  and  D,  invested  $8800  in  real  estate. 
B  invested  twice  as  much  as  A,  C  twice  as  much  as  B,  and  D 
one  half  the  difference  between  B's  and  C's  investments.  Find 
the  capital  invested  by  C. 

8.  Three  men,  A,  B,  and  C,  enter  into  partnership.  A 
furnishes  three  times  as  much  capital  as  B,  and  C  one  half  as 
much* as  A  and  B  together.  The  total  amount  invested  is 
$  7200.     Find  the  amount  invested  by  B. 

9.  A  farmer  has  343  sheep  distributed  in  three  fields.  In 
the  first  field  are  half  as  many  as  in  the  second,  and  in  the 


INTRODUCTION  3 

third  four  times  as  many  as  in  the  first.     How  many  sheep  are 
there  in  each  field  ? 

10.  In  a  room  are  27  persons.  The  number  of  women  is 
twice  the  number  of  men,  and  the  number  of  children  is  twice 
the  number  of  adults.     How  many  are  there  of  each  ? 

11.  Three  boys  divided  110  marbles  among  themselves  so 
that  the  first  of  them  received  twice  as  many  as  the  second, 
and  the  second  three  times  as  many  as  the  third.  How  many 
did  each  receive  ? 

12.  Divide  120  into  two  parts  so  that  one  part  will  be 
5  times  the  other. 

13.  The  difference  between  two  numbers  is  12,  and  the 
greater  is  four  times  the  smaller.     Find  the  numbers. 

Hint.    Let  x  =  the  smaller  number, 

then  4x  =  the  greater  number, 

and  3  X  =  the  difference  between  the  numbers. 

14.  John  has  three  times  as  much  money  as  Henry,  and 
John  has  40  cents  more  than  Henry.     Find  the  share  of  each. 

15.  A  is  three  times  as  old  as  B,  and  A  is  ten  years  older 
than  B.     Find  B's  age. 

16.  A  man  bought  a  horse,  a  cow,  and  a  sheep  for  $133. 
The  price  of  the  cow  was  one  half  the  price  of  the  horse,  and 
the  price  of  the  sheep  one  sixth  the  price  of  the  cow.  Find 
the  price  of  each. 

17.  Find  two  numbers  whose  difference  is  12,  and  one  of 
which  is  five  times  the  other. 

18.  A  line  27  inches  long  is  divided  into  two  parts,  one  of 
which  is  equal  to  eight  times  the  other.  How  long  are  the 
parts  ? 

19.  A  travels  twice  as  fast  as  B,  and  the  sum  of  the  dis- 
tances traveled  by  the  two  is  81  miles.  How  many  miles 
does  each  travel  ? 


4  ELEMENTARY  ALGEBRA 

20.  Two  men  start  at  the  same  time  to  travel,  one  from  A 
to  B,  the  other  from  B  to  A,  two  stations  66  miles  apart.  If 
the  first  man  travels  twice  as  fast  as  the  second,  how  many 
miles  will  he  travel  before  he  meets  the  other? 

NEGATIVE  NUMBERS 
EXERCISE  2 

1.  Subtract  10  from  17. 

2.  Can  10  be  subtracted  from  7  ? 

3.  In  arithmetic  why  cannot  10  be  subtracted  from  7  ? 

4.  The  temperature  at  noon  is  17°  and  at  4  p.m.  it  is  10° 
less.  What  is  the  temperature  at  4  p.m.?  State  this  as  an 
example  of  subtraction. 

5.  The  temperature  at  4  p.m.  is  7°,  and  at  10  p.m.  it  is  10° 
less.     What  is  the  temperature  at  10  p.m.  ? 

6.  Do  you  know  of  any  other  way  of  expressing  the  last 
answer  (3°  below  zero)  ? 

7.  What  then  is  7  -  10  ? 

8.  Can  you  think  of  any  other  practical  examples  which 
require  the  subtraction  of  a  greater  number  from  a  smaller 
one?  

8.  Many  practical  examples  require  the  subtraction  of  a 
greater  number  from  a  smaller  one,  and  in  order  to  express  in 
a  convenient  form  the  results  of  these,  and  similar  examples, 
it  becomes  necessary  to  enlarge  our  concept  of  number,  so  as 
to  include  numbers  less  than  zero. 

9.  Negative  numbers  are  numbers  smaller  than  zero;  they 
are  denoted  by  a  prefixed  minus  sign;  as  —5  (read  ''minus  5"). 
Numbers  greater  than  zero,  for  the  sake  of  distinction,  are  fre- 
quently called  positive  numbers,  and  are  written  either  with  a 
prefixed  plus  sign,  or  without  any  prefixed  sign;  as  -f  5  or  5. 


INTRODUCTION  6 

The  fact  that  a  thermometer  falling  10°  from  7°  indicates  3° 
below  zero  may  now  be  expressed 

7°  _  10°  =  -  3°. 
Instead  of  saying  a  gain  of  $30,  and  a  loss  of  $  90  is  equal  to  a 
loss  of  $  60,  we  may  write 

$30-$90  =  -$60. 

10.   The  absolute  value  of  a  number  is  the  number  taken 

without  regard  to  its  sign. 
+  5 

The  absolute  value  of  —  5  is  5,  of  +  3  is  3. 

+  4 

^3  11.   It  is  convenient  for  many  discussions,  to  represent 

2  the  positive  numbers  by  a  succession  of  equal  distances 
laid  off  on  a  line  from  a  point  0,  and  the  negative  num- 
bers by  a  similar  series  in  the  opposite  direction. 

In  the  annexed  diagram,  e.g.  an  addition  of  3  is  equivalent  to 
-1  an  upward  motion  of  three  spaces ;  hence  3  added  to  —  4  equals 
._2        —  1?  3  added  to  —  1  equals  +  2,  etc. 

Similarly  a  subtraction  of  2  is  equal  to  a  downward  motion  of 
two  spaces.     Hence  2  subtracted  from  0  equals  —  2,  2  subtracted 
-4       from  —  2  equals  —  4,  etc. 

EXERCISE  3 

1.  If  in  financial  transactions  we  indicate  a  man's  income 
by  a  positive  sign,  what  does  a  negative  sign  indicate  ? 

2.  State  in  what  manner  the  positive  and  negative  signs 
may  be  used  to  indicate  north  and  south  latitude,  east  and  west 
longitude,  motion  upstream  and  downstream. 

3.  If  south  latitude  is  indicated  by  a  positive  sign,  by 
what  is  north  latitude  represented  ? 

4.  What  is  the  meaning  of  —  5°  north  latitude  ?  of  the 
year  —  50  a.d.  ?  of  an  easterly  motion  of  —  6  yards  per  second? 

5.  If  the  temperature  at  4  a.m.  is  —  7°  and  at  0  a.m.  it  is  6° 
higher,  what  is  the  temperature  at  9  a.m.?  What,  therefore, 
is  -7  +  6? 


-3 


6  ELEMENTARY  ALGEBRA 

6.  A  vessel  starts  from  a  point  in  25°  north  latitude,  and 
sails  38°  due  south,  (a)  Find  the  latitude  at  the  end  of  the 
journey,     (b)  Find  25  -  38. 

7.  A  vessel  starts  from  a  point  in  25°  south  latitude,  and 
sails  12°  due  south,  (a)  Find  the  latitude  at  the  end  of 
the  journey.     (6)  Subtract  12  from  —  25. 

8.  From  20  subtract  30.  18.   To  -  1  add  2. 

9.  From  4  subtract  6.  19.   From  1  subtract  2. 

10.  From  7  subtract  8.  20.  To  -  7  add  8. 

11.  From  19  subtract  24.  21.  To  -  7  add  2. 

12.  From  0  subtract  12.  22.  From  —  1  subtract  2. 

13.  From  —12  subtract  10.  23.  Add  —1  and  2. 

14.  From  —  2  subtract  4.  Which  is  the  greater  number : 

15.  From  —1  subtract  1.  24.  1  or  —1? 

16.  To -6  add  12.  25.  -lor  -3? 

17.  To  -2  add  1.  26.  -3  or  -4? 

27.  By  how  much  is  —  9  greater  than  —  11? 

28.  What  is  the  difference  in  temperature  between  20°  above 
zero  and  20°  below  zero  ? 

29.  What  is  the  difference  between  +  20  and  —  20  ? 

NUMBERS  REPRESENTED  BY  LETTERS 

12.  For  many  purposes  of  arithmetic  it  is  advantageous  to 
express  numbers  by  letters. 

I.  By  denoting  an  unknown  number  by  x,  it  becomes  possi- 
ble to  perform  arithmetical  operations  with  unknown  quan- 
tities, and  to  solve  problems  thereby  (§  6). 

II.  By  such  a  use  of  letters,  arithmetical  rules  and  principles 
may  be  expressed  very  concisely,  e.g.  if  we  wish  to  express 


INTRODUCTION  7 

briefly  the  principle ;  The  square  root  of  a  fraction  is  equal  to 
the  square  root  of  its  numerator  divided  by  the  square  root  of 
its  denominator,  we  may  write 


4. 


numerator       V  numerator 


/! 


denominator      Vdenominator 
By  representing  the  numerator  by  a  letter,  e.g.  a,  and  the  de- 
nominator by  b,  we  may  write  more  briefly 

b~Vb' 
In  this  equation  a  and  b  denote  any  numbers  whatsoever. 

III.  The  discovery  and  demonstration  of  general  arithmeti- 
cal laws  constitute  the  third  advantage  derived  from  the  em- 
ployment of  letters. 

When  each  of  two  numbers  is  multiplied  by  itself,  and  the 
smaller  product  is  subtracted  from  the  greater,  the  answer  is 
the  same  as  if  we  multiply  the  sum  of  the  two  numbers  by 
their  difference. 

E.g.     (14  X  14)- (13  X  13)  =  196 -169  =  27. 
(14  +  13)  X  (14  -  13)  =    27  X      1  =  27. 

If  we  try  a  great  many  arithmetical  numbers,  we  find  the 
principle  always  correct,  but  this  does  not  prove  that  it  is 
correct  for  all  numbers.  In  algebra,  however,  we  can  easily 
show  that  (a  X  a)  —  (6  X  6)  =  (a  +  6)  X  (a  —  b)  (§  65),  and  since 
a  and  b  denote  any  number  whatsoever,  the  law  is  correct  for 
any  two  numbers. 

IV.  For  a  fourth  advantage,  see  §  30. 

EXERCISE  4 

1.  If  the  letter  a  means  1000,  what  is  the  value  of  6  a  ? 

2.  What  is  the  value  of  6a  if  a  =  7?   ifa  =  i? 

3.  If  a  boy  has  6d  marbles  and  wins  4cZ  marbles,  how 
many  marbles  has  he  ? 


8  ELEMENTARY  ALGEBRA 

4.  Is  the  last  answer  correct  for  any  value  of  d  ? 

5.  What  is  the  sum  of  9  6  and  6  &  ? 

6.  Find  the  numerical  value  of  the  last  answer  if  6  =  17. 

7.  If  c  represents  a  certain  number,  what  represents  10 
times  that  number? 

8.  From  24  m  subtract  15  m. 

9.  What  is  the  numerical  value  of  the  last  answer  if  m=7  ? 

10.  From  12  m  subtract  15  m,  and  find  the  numerical  value 
of  the  answer  if  m  =  12. 

11.  Add  12p,  2p,  1  p,  and  subtract  24p  from  the  sum. 

12.  From  —  12  g  subtract  20  q. 

13.  Add  —  12g  and  +12^'. 

14.  From  0  subtract  12  x. 

15.  Add  —  3  a;  and  6  x. 

16.  From  — 12  a;  subtract  0. 

17.  From  — 12^9  subtract  12^. 

18.  If  a  =  100,  then  7a  =  700.     What  sign,  therefore,  is 
understood  between  7  and  a  in  the  expression  7a? 

FACTORS,  POWERS,   AND  ROOTS 

13.  If  there  is  no  sign  between  two  letters,  or  a  letter  and  a 
number,  a  sign  of  multiplication  is  understood. 

5  X  a  is  generally  written  5  d,  w  x  w  is  written  mn. 
Between  two    figures,   however,   a   sign   of    multiplication 
(either  x  or  •)  has  to  be  employed ;  as,  4  x  7,  or  4  •  7. 
4x7  cannot  be  written  47,  for  47  means  40  +  7. 

14.  A  product  is  the  result  obtained  by  multiplying  together 
two  or  more  quantities,  each  of  which  is  a  factor  of  the  product. 

Since  24  =  3  x  8,  or  12  x  2,  each  of  these  numbers  is  a  factor  of  24. 
Similarly,  7,  a,  6,  and  c  are  factors  of  7  ahc. 


INTRODUCTION  9 

15.  A  power  is  the  product  of  two  or  more  equal  factors ; 
thus,  aaaaa  is  called  the  "  fifth  power  of  a,"  and  written  a* ; 
aaaaaa,  or  a^,  is  "the  6fch  power  of  a,"  or  a 6th. 

The  second  power  is  also  called  the  square,  and  the  third 
power  the  cube;  thus,  12^  (read  "  12  square")  equals  144. 

16.  The  base  of  a  power  is  the  number  which  is  repeated  as 
a  factor. 

The  base  of  a^  is  a. 

17.  An  exponent  is  the  number  which  indicates  how  many- 
times  a  base  is  to  be  used  as  a  factor.  It  is  placed  a  little 
above  and  to  the  right  of  the  base. 

The  exponent  of  m^  is  6  ;  n  is  the  exponent  of  a". 

EXERCISE  5 

1.  Write  and  find  the  numerical  value  of  the  square  of  6, 
the  cube  of  7,  the  fourth  power  of  2. 

Find  the  numerical  values  of  the  following  powers : 

2.  61  5.    1^0.  8.    0^  11.    20\  14.    1.21 

3.  2«.  6.    2\  9.    (21)2.        12.    (if.  15.    ay. 

4.  10^.         7.    S\         10.    3.51  13.    .0013. 

If  a  =  2,  b  =  3,c  =  l,  and  d  =  l,  find  the  numerical  values  of : 

16.  a^         18.    c\  20.    d'^'.  22.    5*.  24.    a\ 

17.  b\          19.    d\  21.    5\  23.    5^  25.  -ft'*. 

26.  The  distance  of  the  north  pole  from  the  equator  is  10' 
meters.     Find  the  distance  in  meters. 

27.  A  kilometer  equals  10  hectometers,  1  hectometer  =  100 
meters,  1  meter  =  100  centimeters,  and  1  centimeter  =  10 
millimeters. 


10  ELEMENTARY  ALGEBRA 

Express  as  a  power  of  10  the  number  of 

(a)  Millimeters  contained  in  a  meter. 

(5)  Millimeters  contained  in  a  kilometer, 

(c)  Square  millimeters  contained  in  a  square  kilometer. 

28.  Express  as  a  power  of  2  the  number  of  parents,  grand- 
parents, great-grandparents,  etc.,  a  person  may  have. 

29.  What  exponent  is  understood  when  none  is  written,  as 
in  X  ov  y? 


18.  In  a  product  any  factor  is  called  the  coefficient  of  the 
product  of  the  other  factors. 

In  12  mn^p,  12  is  the  coefficient  of  mn^p,  12  m  is  the  coefficient  of  n^p. 

19.  A  numerical  coefficient  is  a  coefficient  expressed  entirely 
in  figures. 

In  —  17  xyz,  —  17  is  the  numerical  coefficient. 

When  a  product  contains  no  numerical  coefficient,  1  is 
understood ;  thus  a  =  1  a,  cc^b  =  1  o?b. 

20.  When  several  powers  are  multiplied,  the  beginner  should 
remember  that  every  exponent  refers  only  to  the  number  near 
which  it  is  placed. 

3  a^  means  3  aa,  while  (3  a)^  =  3  a  x  3  a. 
9  ahy^  —  9  ahyyy. 
16  x'^y^z  =  16  xxyyyz. 

EXERCISE  6 
If  a  =  2,  6  =  1,  c  =  3,  and  a;  =  4,  find  the  numerical  value  of : 

1.  3  a.  4.    a\  7.  ^a\  10.    1  ah\ 

2.  ahc.  5.    620,  ^     557,  ^^     ^^^^^ 

3.  6  6a;.  6.    4a6ca;.  9.    9  0^.  12.    'da'W. 


INTRODUCTION  11 

If  a  =  3,  6  =  2,  c  =  1,  c?  =  4,  find  the  numerical  value  of : 

13.  14a&V.        16.    10  ab^d.        19.    2^  22.    ^a^^. 

14.  Ta^^V.         17.    a^  20.    7".  23.    a^ 

15.  8a6c^.  18.    4'*.  21.    fa^  24.    &^ 

If  a  =  2,  2>  =  3,  g  =  4,  a;  =  1^,  2/  =  0,  find  the  numerical  value  of : 

25.  3gpV.  29.   a^.  33.    2«a^. 

26.  a'^p^y.  on     ^ 

27.  ^-  31.    4^.  32 

35.    —  • 

28.  ay.  32.    4a2px2.  a« 

36.  What  is  the  numerical  coefficient  in  each  of  the  expres- 
sions in  Exs.  1-6  ? 

37.  What  are  the  coefficients  of  x  in  Exs.  3  and  6  ? 


21.  A  root  is  one  of  the  equal  factors  of  a  power.  According 
to  the  number  of  equal  factors,  it  is  called  a  square  root,  a 
cube  root,  a  fourth  root,  etc. 

.S  is  the  square  root  of  9,  for  3^  =  9. 
5  is  the  cube  root  of  125,  for  5^  =  125. 
a  is  the  fifth  root  of  a^,  the  nth  root  of  «»*. 

The  Tith  root  is  indicated  by  the  symbol  •>/  ;  thus  V«  is 
the  fifth  root  of  a,  V27  is  the  cube  root  of  27,  V«,  or  more 
simply  Va  is  the  square  root  of  a. 

Using  this  symbol  we  may  express  the  definition  of  root  by 

22.  The  index  of  a  root  is  the  number  which  indicates  what 
root  is  to  be  taken.  It  is  written  in  the  opening  of  the  radical 
sign. 

In  Va,  7  is  the  index  of  the  root. 


14.    (a-\'b)Vb. 


12  ELEMENTARY  ALGEBRA 

23.  The  signs  of  aggregation  are :  the  parenthesis,  (  ) ;  the 
bracket,  [  ] ;  the  brace,  {  j ;  and  the  vinculum, . 

They  are  used,  as  in  arithmetic,  to  indicate  that  the  expressions 
included  are  to  be  treated  as  a  whole. 

Each  of  the  forms  10  x  (4  +  1),  10  x  [4  +  1],  10  x  4  +  1  indicates  that 
10  is  to  be  multiplied  by  4  +  1  or  by  5. 

(a  —  b)  is  sometimes  read  "  quantity  a  —  6." 

EXERCISE  7 
If  a  =  2,  6  =  4,  c  =  1,  d  =  0,  x  =  9,  find  the  numerical  value  of : 

1.  -Vb.  7.    VP.  13.    3(a  +  &). 

2.  -^la.  8.    a/^. 

3.  V45.  9.    aV6. 

4.  </3x.  '      10.    2aV9b.  ''•    f^  +  ^^J^^' 

5.  V4^.  11.    4.adV5rb.  16-    («  +  ^)'- 

6.  V9c.  12.    a  +  &.  17.    (b  +  cf. 

18.    In  Ex.  14  what  is  the  coefficient  of  the  V6?  of  (a  +  6)V6? 


ALGEBRAIC   EXPRESSIONS  AND  NUMERICAL 
SUBSTITUTIONS 

24.  An  algebraic  expression  is  a  collection  of  algebraic  sym- 
bols representing  some  number ;  e.g.  6  a^b  —  7  Vac^  -h  9. 

25.  A  monomial  or  term  is  an  expression  whose  parts  are  not 

separated  by  a  sign  +  or  —  ;  as  3  ax%  —  9  Va;,  ~    ^   . 

o  c 

«(&  +  c  4-  cZ)  is  a  monomial,  since  the  parts  are  a  and  (6  +  c  +  d). 

26.  A  polynomial  is  an  expression  containing  more  than  one 

term. 

7  X        /- 
4x4-2/,  -^  +  v^  —  3  a^6  ;  and  a*  +  &*  4-  c*  +  (Z*  are  polynomials. 


INTRODUCTION  13 

27.  A  binomial  is  a  polynomial  of  two  terms. 
d?'  +  6^,  and  |  —  Va  are  binomials. 

28.  A  trinomial  is  a  polynomial  of  three  terms. 
a-\-h  +  c^  a  +  964-  VS  are  trinomials. 

29.  In  a  polynomial  each  terra  is  treated  as  if  it  were  con- 
tained in  a  parenthesis,  i.e.  each  term  has  to  be  computed 
before  the  different  terms  are  added  and  subtracted.  Otherwise 
all  operations  of  addition,  subtraction,  multiplication,  and  divi- 
sion are  to  be  performed  in  the  order  in  which  they  are  written 
from  left  to  right. 

E.g.  3  +  4-5  means  3  +  20  or  23. 

Ex.  1.    Find  the  value  of  4  •  2^  +  5  •  3^-  ^^. 

4.23  +  5.32-^^ 
2 

=  4.8  +  5.9-  — 

=  32  +  45-27 

=  50. 

Ex.  2.  If  a  =  5,  6  =  3,  c  =  2,  d  =  0,  find  the  numerical  value 
of  6  a62  _  9  aWc  -[-^a^b- 19  a^hcd. 

6  a&2  _  9  «62c  + 1  a35  _  19  a%cd 
=  6  .  5  .  32  -  9  .  6  .  32  .  2  +  1 .  53 .  3  -  19  .  52  .  3  .  2  . 0 
=  6. 5. 9-9. 5.9.2  +  1. 125. 3-0 
=  270  -  810  +  150 
=  -390. 

EXERCISE  8 

1.  State  what  kind  of  expressions  are  Exs.  18-27  of  this 
exercise. 

If  a=:5,  h  =  2,   c  =  l,  (Z  =  0,  x  =  \,  and  y^\,   find  the  nu- 
merical value  of: 

2.  a  +  6  +  3c.>  4.   a  — &. 

3.  a  -f-  5  6c.  5.   a  —  4  6. 


14  ELEMENTARY  ALGEBRA 

6.  5 a- -\- Tab.  18.  (a-^b)c. 

7.  ah-2hh-4.cd.  19.  a'^x(h  +  2c). 

8.  ^ahc  — 20 he- ^Oabd.  20.  a  +  Va^  +  11  c. 

9.  7a  +  12cZ-19c-206.  21.  2a3_  (^.^  5)(c  +  d). 

10.  5a6+6c2-4MH-7a2.  22.   30.T2/(a4-Z>)Va  +  6+c+2a;. 

11.  4c  +  6a-8&^H6a;.  __     a  ,  6  ,   c 

^^'    K  +  o  +  Q- 

12.  4.cx  +  lh^-2a'-4.x.  b      Z     6^ 

13.  a«  +  63  +  c«  +  d«.  24.   9a6c  +  Vc  +  4c7. 

14.  3ad-66^  +  2dc.  +  45l         "'•    K«  +  ^) +cjd-6a.2/. 

15.  2a26-6a//  +  7ac2-9aU   26.   ^+1^  +  ?. 

'  2c       4       2/ 

17.    2x2  +  3/.  a^6^c 

28.7  c-d2  -  aj(3  ahx  +  r)  -  c.r(d  +  h)-12  (?x. 

29.  ^+^.  31.    ^^±1.  33.    l^^±li^. 
a-6                             a-^-9  S^^  +  Sc^ 

30.  l_l  +  i.  32.    ^4+51. 
a      h      c  o? -\-}r 

34.    (a  +  6)  (c  +  d)  -  (a  +  c)  (6  +  (^)  +  (a  +  (^)(6  -  c). 

Express  in  algebraic  symbols : 

35.  Six  times  a  plus  3  times  h. 

36.  Six  times  the  square  of  a  minus  five  times  the  cube  of  6. 

37.  Eight  X  cube  minus  six  x  cube  plus  y  square. 

38.  Six  m  cube  plus  four  times  the  quantity  a  minus  h. 

39.  The  quantity  a  plus  6  multiplied  by  the  quantity  c^ 
minus  V^. 

40.  Twice  o?  diminished  by  5  times  the  square  root  of  the 
quantity  a  minus  h  square. 

41.  Read  the  expressions  of  Exs.  2-0^  of  the  exercise. 


INTRODUCTION  15 

30.  The  representation  of  numbers  by  letters  makes  it  pos- 
sible to  state  very  briefly  and  accurately  some  of  the  principles 
of  arithmetic,  geometry,  physics,  and  other  sciences. 

Ex.     If  the  three  sides  of  a  triangle  contain  respectively 

a,  b,  and  c  feet  (or  other  units  of  length),  and  the  area  of  the 

triangle  is  S  square  feet  (or  squares  of  other  units  selected), 

then  , 

S  =  i  V  (a  +  6  +  c)  (a  +  6  —  c)  (a  —  6  +  c)  (6  —  a  +  c). 

E.g.  the  three  sides  of  a  triangle  are  respectively  13,  14,  and 
15  feet,  then  a  =  13,  6  =  14,  and  c  =  15 ;  therefore 

^=iV(13+14+15)(13+14-15)  (13-14+15)  (14-13+15) 
=  ^  V42  .  12  .  14  .  16 

=  ix336  „ 

=  84,  i.e.  the  area  of  the  triangle  equals 
84  square  feet. 

EXERCISE  9 
1.    By  using  the  formula 


S  =  ^V(a  +  6  +  c)  (a  +  6  —  c)  (a  —  6  +  c)  (6—  a  +  c), 

find  the  area  of  a  triangle  whose  sides  are  respectively 

(a)  5,  12,  and  13  feet.  (c)  4,  13,  and  15  meters. 

(&)  3,  4,  and  5  inches.     '  (d)  9,  10,  and  17  yards. 

2.  If  the  radius  of  a  circle  is  B  units  of  length  (inches, 
meters,  etc.),  the  area  S  =  3.1416  •  M^  square  units  (square 
inches,  square  meters,  etc.).  Find  the  area  of  a  circle  whose 
radius  is 

(a)  1000  meters.        (6)  3  inches.  (c)  240,000  miles. 

3.  If  i  represents  the  simple  interest  of  p  dollar;^  at  r%  in 
n  years,  then  i  =p  •  n  •  r  %,  or  y^« 


16  ELEMENTARY  ALGEBRA 

Mnd  by  means  of  this  formula  : 

(a)  The  interest  on  $730  for  4  years  at  21%. 

(b)  The  interest  on  $380  for  2  years  at  4%. 

(c)  The  interest  on  $246  for  4  months  at  7%. 

4.  If  /represents  the  compound  interest  of  p  dollars  at  r% 
for  n  years  (compounded  yearly),  then  ^=P[^~^jw-A   ~ P- 

Find  the  compound  interest  of  ; 

(a)  $400  for  3  years  at  10%. 
(6)   $  1200  for  4  years  at  20%. 
(c)    $1  for  2  years  at  5%. 

5.  If  the  diameter  of  a  sphere  equals  d  units  of  length,  the 
surface  S  =  3.1416  d?  (square  units).  (The  number  3.1416  is 
frequently  denoted  by  the  Greek  letter  tt.  This  number  can- 
not be  expressed  exactly,  and  the  value  given  above  is  only  an 
approximation.) 

Find  the  surface  of  a  sphere  whose  diameter  equals  : 

(a)  8000  miles.  {h)  2  inches.  (c)    12  feet. 

6.  If  the  diameter  of  a  sphere  equals  d  feet,  then  the  volume 

F=— cubic  feet. 
6 

Find  the  volume  of  a  sphere  whose  diameter  equals : 
(a)  10  feet.  (6)   6  feet.  (c)   11  feet. 

7.  A  body  falling  from  a  state  of  rest,  passes  in  t  seconds 
over  a  space  S  =■  \  gt^.  The  value  of  g  for  New  York  is  32.16 
feet,  or  980  cm.  (This  formula  does  not  take  into  account  the 
resistance  of  the  atmosphere.) 

(a)  How  far  does  a  body  fall  from  a  state  of  rest  in  3 
seconds? 

(6)  A  stone  dropped  from  the  top  of  a  tree  reached  the 
ground  in  2^  seconds.     Find  the  height  of  the  tree. 


INTRODUCTION  17 

(c)  How  far  does  a  body  fall  from  a  state  of  rest  in  J^j  of  a 
second  ? 

(d)  On  the  surface  of  the  moon,  how  far  does  a  body  fall 
from  a  state  of  rest  in  3  seconds,  if  ^  =  5.4  ft.  ? 

8.  If  i^  denotes  the  number  of  degrees  of  temperature  indi- 
cated on  the  Fahrenheit  scale,  the  equivalent  reading  G  on  the 
Centigrade  scale  may  be  found  by  the  formula 

(7  =  |(P-32). 

Change  the  following  readings  to  Centigrade  readings : 
(a)  100°  F.  (6)  32°  F.  (c)  5°F. 


CHAPTER   II 

ADDITION,   SUBTRACTION,   AND   PARENTHESES 

ADDITION   OF   MONOMIALS 

31.  While  in  arithinetic  the  word  sum  refers  only  to  the 
result  obtained  by  adding  positive  numbers,  in  algebra  this 
word  includes  also  the  results  obtained  by  adding  negative,  or 
positive  and  negative  numbers. 

In  arithmetic  we  add  a  gain  of  $  6  and  a  gain  of  $  4,  but  we 
cannot  add  a  gain  of  $  6  and  a  loss  of  $  4.  In  algebra,  how- 
ever, we  call  the  aggregate  value  of  a  gain  of  6  and  a  loss  of  4 
the  sum  of  the  two.  Thus  a  gain  of  $2  is  considered  the  sum 
of  a  gain  of  $  6  and  a  loss  of  $  4.     Or  in  the  symbols  of  algebra 

(+$6) +  (-$4)  =+$2. 

Similarly,  the  fact  that  a  loss  of  $  6  and  a  gain  of  $  4  equals  a 
loss  of  $  2,  may  be  represented  thus 

(-$6) +  (+$4)  =  (-$2). 

In  a  correspdnding  manner  we  have  for  a  loss  of  ^6  and  a  loss 

of  $4  (_$6)4-(-|4)  =  (-$10). 

Since  similar  operations  with  different  units  always  produce 
analogous  results,  we  define  the  sum  of  two  numbers  in  such 
a  way  that  these  results  become  general,  or  that 

6+(-4)  =  +  2, 

(_6)  +  (+4)  =  -2, 

(_6)  +  (-4)  =  -10, 

and  (+6)  +  (+4)  =  +  10. 

18 


ADDITION,   SUBTRACTION,  AND  PARENTHESES     19 

32.    These  considerations  lead  to  the  definition  of  addition : 

If  two  numbers  have  the  same  sign,  add  their  absolute  values; 
if  they  have  opposite  signs,  subtract  their  absolute  values  and 
(always)  prefix  the  sign  of  the  greater. 


Find  the  sum  of : 


EXERCISE   10 


1.      +6  2.      -6  3.      -3  4.      +3 

-3  +3  -6  +6 

5.-7  6.      +7  7.        11  8.    -11 

+  8  -8  -   4  -   4 


9. 


9 

10. 

-20 

11. 

0 

12. 

0 

-9 

-12 

7 

-7 

6 

14. 

-1 

15. 

1 

16. 

-17 

6 

-2 

—  2 

12 

-11 

-3 

±i 

+   5 

13 


Find  the  value  of : 

17.  (-20) +  (-21).  20.  0  +  (-18). 

18.  (-12)  +  13.  21.  (-l)+2+(-3)  +  4. 

19.  l+(-7).  22.  (-6)  +  (-7)4-(-8)  +  (-9). 

In  Exs.  23-26,  find  the  numerical  value  of  a  +  b  +  c  +  d,  it: 

23.  a  =  l,  6  =  -2,  c  =  -4,  d  =  6. 

24.  a  =  7,b  =  -7,c  =  0,d  =  l. 

25.  a=-7,  6  =  8,  c  =  -9,  d  =  10. 

26.  a  =  17,  6  =  12,  c  =  - 17,  d  =  - 11. 

27.  What  number  must  be  added  to  3  to  give  7  ? 

28.  What  number  must  be  added  to  7  to  give  3  ? 


20  ELEMENTARY  ALGEBRA 

29.  What  number  must  be  added  to  —  7  to  give  3  ? 

30.  What  number  must  be  added  to  —  3  to  give  —  7  ? 

31.  Add  4  yards,  7  yards,  and  3  yards. 

32.  Add  4  a,  7  a,  and  3  a. 

33.  Add  4  a^bx,  7  a^bx,  and  —  3  a^bx. 


33.   Similar  or  like  terms  are  terms  which  have  the  same 
literal  factors,  affected  by  the  same  exponents. 


a'^b 


6  ax'^y  and  -  7  ax'^y,  or  —  5  a'^b  and  ^,  or  16  Va  -\- b  and  -  2\/«  +  6, 
are  similar  terms.  ' 

Dissimilar  or  unlike  terms  are  terms  which  are  not  similar. 

4  a^bc  and  —  4  a^bc^  are  dissimilar  terms. 

34.  The  sum  of  two  similar  terms  is  another  similar  term. 
The  sum  of  3  x"^  and  -  |  x^  is  |  x^. 

Dissimilar  terms  cannot  be  united  into  a  single  term.  The 
sum  of  two  such  terms  can  only  be  indicated  by  connecting 
them  with  the  +  sign. 

The  sum  of  a  and  a^  is  a  +  d^. 

The  sum  of  a  and  —  6  is  «  +  (—  6),  or  a  —  &. 

35.  Algebraic  sum.  In  algebra  the  word  sum  is  used  in  a 
wider  sense  than  in  arithmetic.  While  in  arithmetic  a  —  b 
denotes  a  difference  only,  in  algebra  it  may  be  considered 
either  the  difference  of  a  and  b  or  the  sum  of  a  and  —  b. 

The  sum  of  —  a,  —  2  a&,  and  4  ac^  is  —  a  —  2  a6  +  4  adK 


Add: 

EXERCISE   11 

..    -3a 

2. 

a^            3.        7  :f?y 

4. 

—  4  mr? 

-4a 

-2a2                  -l:»?y 

—  5  mr? 

-5a 

+  3a2                  +    :^y 

—  7  m?i* 

ADDITION,   SUBTRACTION,   AND  PARENTHESES     21 

7.        7  mnp  9.    14  aha? 

15  ah:i? 
-17  a&a^ 


5. 

12maJ2 

15mxz 

- 

-28  maj2 

6. 

c^de 

-    c^de 

Ac'de 

7  mnp 

—  8  mup 

—  9  mnp 

.       8  a^ftc^ 

-   da'bc" 

11  a^Z^c^ 

8.       8a2?>c2  10.    lla^fe^ 

-  21  a'b' 


Find  the  sum  of : 

11.  6  a,  7  a,  —2  a,  — 12  a,  — 15  a. 

12.  —  7  iC2/j  12  a??/,  —  19  xy,  —  7  xy. 

13.  -  8  ah&,  -  7  ahc\  -  9  ahc\  -  ah(?. 

14.  4(a  +  &),  -  5(a  +  6),  6(a  +  6),  -  7(a  +  h). 


15.    -6V^T^,  -7Va;  +  2/,  -8Vx4-2/,  -9Va;4-2/. 


16.  4Va  +  6  +  c«,  -8Va  +  6  +  c^,  -7Va  +  &4-c«. 
Find  the  value  of : 

17.  —llxy  —  19  xy  —  xy  —  27  xy  —  30  a?!/- 

18.  15m  +  18  7/i  + w  — 7  m  — 10  m  — 14  m. 

19.  _jnL2j2_822_1^^2_|_42!2. 

20.  5mw  — 6m7t-f  7mn  — 9mn  +  12mn. 

21.  \ah^-lah''-^^ay'^-%ab\ 

22.  1.2x2y2  +  2.4a^2/^  +  .7a.'2?/;3-1.3a^?/2?. 

23.  fp'g-ffp'g-fpV 

24.  -1-  a;y;s2  _,_  ^  3522^2^2  _  |  ^2^2^2  _  ^^^2^2^ 

Add: 

25.    a  26.        a  21.    —a  28.        o 

h  -b  —b  -a* 


29.    a  30.        a^  31.        a^  32.    -2 

1  -1  -6  b 


22 

—  ax 

-ax' 

ELEMENTARY  ALGEBRA 

34.    4:abc             35.    ab 
4  ab'c                     0 

36. 

33. 

-Va-& 

Simplify  the  following  by  uniting  like  terms : 

37.  6a-{-4:b  —  5a-15b-\-2a  +  4ta  —  7b  +  b. 

38.  10  c  — 11  m-{-5x  —  4:y  —  4:X  — 12  c-\-y  +  x  —  m. 

39.  12  mn^  — 12  mn^  —  11  mn^  + 11  m^^  —  17  a;. 


41.    6  ic^2/ "~  1'^  ^y  ~  19  ^^  + 12  *^^^  ~  ^  ^^^ "~  ^V' 

ADDITION  OF  POLYNOMIALS 

36.  Polynomials  are  added  by  uniting  their  like  terms.  It 
is  convenient  to  arrange  the  expressions  so  that  like  terms  may 
be  in  the  same  vertical  column,  and  to  add  each  column. 

Thus,  to   add   26  ab  -  S  abc  -  15  be,  - 12  ab -\- 15  abc  -  20  c% 

—  5  ab-{- 10  be  — 6  (f,   and    —  7  abc  +  4  6c  +  c^,   we    proceed    as 
follows:  26  ab-   S  abc -15  be 

-  12  ab  +  15  ahc  -  20  c^ 

-  5ab  +  10  &c  -    6  c2 
-    7a&c+    4&C+      c^ 

9ab  -      6c-25c2    Sum. 

37.  Numerical  substitution  offers  a  convenient  method  for 
checking  the  sum  of  an  addition.     To  check  the  addition  of 

—  3a  +  46  +  5c  and    +2a  — 26  — c    assign    any    convenient 
numerical  values  to  a,  b,  and  c,  e.g.  a  =  1,  6  =  2,  c  =  1, 

then  -3a  +  46  +  5c=-3  +  8  +  5  =  10, 

2a-26-    c=      2-4-l  =  -3, 

the  sum  —    a  +  26  +  4c=— H-4-f-4  =  7. 

But  7  =  10  —  3,  therefore  the  answer  is  correct. 

Note.  While  the  check  is  almost  certain  to  show  any  error,  it  is  not 
an  absolute  test,  e.g.  the  erroneous  answer  — a  +  66— 4c  would  also 
equal  7. 


ADDITION,   SUBTRACTION,   AND  PARENTHESES      23 

38.  In  various  operations  with  polynomials  containing  terms 
with  different  powers  of  the  same  letter,  it  is  convenient  to 
arrange  the  terms  according  to  ascending  or  descending  powers 
of  that  letter. 

7  +  X  -\-  5x^  -\-  7  x^  +  5x^  is  arranged  according  to  ascending  powers 
of  X.  5  a^  _  7  0,%  +  4  a'^bc  -  8  a'^bcP  +  7  qb^d  +  9  6^  +  e^  is  arranged  ac- 
cording to  descending  powers  of  a. 

EXERCISE  12 

Add  the  following  polynomials  : 

1.  5a-6b-7c,  -3a-{-2b-9c,  and  -8a-56  +  llc. 

2.  Sx  —  5y-^7z,5x-\-9y  —  8z,  —4 x—5y-i-S z,  and  —14 x 
-{-6y-z. 

3.  2a2-962_3c2,  - 5a' +  11  b^- 9 c\  4. a^ -3 b^-\-5c^,  said 
-eb'-7c'-\-Sa\ 

4.  5r  —  6x  —  9z-{-ll  v,7r  —  9x  —  llz-]-Sv,  4r  — 82;,  and 
8  a;  + 14  u 

5.  26m+10x  +  Uv+z,  -12m-\-15x-20z,  -12x-5m 

—  5z,  and  11 2;  —  7  ». 

6.  12m  — lip -[-13 z,  —4.m-{-3p-{-y,  7m  —  x  —  5y,  —Sin 
-[-2x  —  y,  and  —  10  m  — 8j9  +  42/. 

7.-5 xy  —  2yz  —  7xz,  Sxy -{-3yz  —  2z%  —2  xy  +  4:XZ-\-5  z'^, 

—  xy  -\-  yz  —  4:.z'. 

8.  3a'-2b'-hc',  -2a2  +  62_3c2,  -  a'-\-3b'-2c',  a'  + 
2  7/H-4c2. 

9.  13(a  +  6)-5(6  +  c)  +  7(c  +  d),  5(a  +  6)  +  9(&  +  c) - 
S(c  +  d),  -4(a  +  &)-5(6  +  c)-f3(c4-<^),  and  -14(a-f-6)- 
(c-\-d)-j-6{b  +  c). 

10.  4:Vx  —  Vy  —  3Vz,  2Vy-\--\/z,  and  —  4V^-Vy  — V«. 

11.  a^-a^  +  2a3,  3a^-4a*H-6  a^,  _8a^-7a^  +  8a3,  ^^^ 
-3a'-9a'-16a\ 


24  ELEMENTABY  ALGEBRA 

12.  1  +  cc  —  a?2,  l  —  x-\-o(?,  —l-\-x-\-x^,  and  —x-\-x^. 

13.  8(c»  +  2/y-7(x  +  2/)  +  6,  -^(x-\-yf-Z{x^y)-b,  and 

14.  a3  +  3a26  +  3a62_|.53^    _2a^-2W,    o?-2o?h-W,    and 
-2a26_&3. 

15.  21pq  —  llxy-\-^'if-,    —  21  pq- 9 y^ -]-xy,    and  y^—pq  + 
ITxy. 

16.  a  — 6j  &  — c,  c  — d,  d  — e,  and  e  — a. 

17.  Ta^  +  U^-d'-^-d^    d'-d'-e^,  b'-a^  +  c^,    and   c^-ft^ 

„3 


-a-^. 


18.  -3.5a -5.7  6  + 1.8  c,   5.3  a  -  4.3  c  -  3.6  5,    11.26-2.2c 
-  7.4  a. 

19.  x'-2x'  +  x^-l,  2a^-2x^  +  l,  2  +  x\ 

20.  i^_22/3-lla;V-a^/,   4:y'-Sa:^  +  2x^y,    Tx^y-6xy^-{- 
a^,  7oif  —  4:xy^-\-4:X^y. 

21.  2a^-a^-a,  4.a^-6a^  +  a,  -a^ -\-Sa^ +  7 a. 

22.  3m34-5m4-8,  10m^-6-4.m\  2m^-2m-3. 

23.  4i)3  +  7/-i)  +  l,    6p2_3y4_p^    7_2p4_|_p2^    -p^  + 
4p  —pi 

24.  b'-b'  +  b'-b-{-l,    -2b'  +  2b'-2b'-2b  +  2,    3b'- 
3b^-j-3b^-3b-hS. 

25.  -9a  +  3  +  16a^  +  a^,  13a2  +  5-4a  +  8a^  lla-15  + 
7a2  +  6al 

26.  5a^-4.a''b-^3ab''-2b%    -4:a^  +  3a^b -2ab^ +  b%    3a^ 
-2a'b-\-ab^,  4.a^  +  3a'b-2ab''  +  b\ 

27.  7a'-4.b'-\-3a^b-2a^b^  +  7ab%   -7  ab^  +  4:a^b-7  a'-\- 
3a'b%  b'-3a?b\ 

28.  Qa'-7b\  3 a^b  +  3 a^b%  6  a^b^ - 5 a^  b'-6 a%. 

29.  ia^-2x^  +  ia;-3,  Jx^-f  a^  +  i    2x-^x^. 

30.  ia;2-ia;-i,  ia^^-Ja;-!    iaj2__i.^_| 


ADDITION,    SUBTRACTION,   AND  PARENTHESES      25 

SUBTRACTION 
EXERCISE   13 

1.  What  is  the  remainder  if  6  is  taken  from  12  ? 

2.  If  from  the  6  negative  units,   —1,  —1,  —1,   —1,  —  1, 

—  1,  four  negative  units  are  taken,  how  many  negative  units 
remain  ?  What  is  therefore  the  remainder  when  —  4  is  taken 
from  —6? 

3.  Instead  of  subtracting  in  the  preceding  example,  what 
number  may  be  added  to  obtain  the  same  result  ? 

4.  The  sum  total  of  the  units  +1,  +1,  +1,  +1?  +1?  —  1? 

—  1,  and  —  1,  is  2.  What  is  the  value  of  the  sum  if  two  nega- 
tive units  are  taken  away  ?  If  three  negative  units  are  taken 
away? 

5.  What  is  therefore  the  remainder  when  —  2  is  taken  from 
2  ?    When  -  3  is  taken  from  2  ? 

6.  What  other  operations  produce  the  same  result  as  the 
subtraction  of  a  negative  number  ? 

7.  If  you  diminish  a  person's  debts,  does  he  thereby  become 
richer  or  poorer  ? 

8.  State  other  practical  examples  which  show  that  the  sub- 
traction of  a  negative  number  is  equal  to  the  addition  of  a 
positive  number. 

39.  Subtraction  is  the  inverse  of  addition.  In  addition,  two 
numbers  are  given,  and  their  algebraic  sum  is  required.  In 
subtraction,  the  algebraic  sum  and  one  of  the  two  numbers  is 
given,  the  other  number  is  required.  The  algebraic  sum  is 
called  the  minuend,  the  given  number  the  subtrahend,  and  the 
required  number  the  difference. 


26  ELEMENTARY  ALGEBRA 

Therefore  any  example  in  subtraction  may  be  stated  in  a  dif- 
ferent form;  e.g.  from  —5  take  —3,  may  be  stated:  What 
number  added  to  —  3  will  give  —  5  ?  To  subtract  from  a  the 
number  h  means  to  find  the  number  which  added  to  b  gives  a. 
Or  in  symbols,  a-h  =  x, 

if  x-{-h  =  a. 

Ex.  1.    From  5  subtract  —3. 

The  number  which  added  to  —  3  gives  5  is  evidently  8. 

Hence,  5 -(-3)  =  8. 

Ex.  2.   From  —  5  subtract  —  3. 

The  number  which  added  to  —  3  gives  —  5  is  —  2. 

Hence,  (_  5)  _(_  3)  =  - 2. 

Ex.  3.     From  —  5  subtract  +  3. 
This  gives  by  the  same  method, 

-5-(+3)  =  -8. 

40.  The  results  of  the  preceding  examples  could  be  obtained 
by  the  following 

Principle.  To  subtract,  change  the  sign  of  the  subtrahend  and 
add. 

The  numerical  results  of  Exs.  1-3  of  course  do  not  prove 
this  principle,  but  it  may  be  deduced  as  follows : 

The  principle  is  obviously  correct  for  a  positive  subtrahend. 

To  find  a  —(—6)  we  have  to  find  the  number  which  added 
to  —  6  will  give  the  result  a. 

But  a  +  6  added  to  —  b,  gives  a. 

Hence  the  required  remainder  is  a-\-bj 

or,  a-(-6)  =  a  +  6. 

Note.  The  student  should  perform  mentally  the  operation  of  chang- 
ing the  sign  of  the  subtrahend  ;  thus  to  subtract  —  8  a^h  from  —  6  a-h, 
change  mentally  the  sign  of  —  8  a^h  and  find  the  sum  of  —  6  a%  and 
-h  8  a-2&. 


ADDITION,   SUBTRACTION,  AND  PARENTHESES     27 

41.   To  subtract  polynomials  we  change  the  sign  of  each  term 
of  the  subtrahend  and  add. 

Ex.   From  -Qa^-Sx^  +  J  subtract  2  a^-Sx^-5x-\-S. 

Check,  If  a;  =  1 

-6a^-3a^4-7  =-2 

2a.-3-3a;2_5a._|_8  ^^2 


-Sx^-\-5x- 

-1 

=  -4 

From 
1.   6  take  8. 

EXERCISE   14 

5.-6  take  —  8. 

9. 

-11  take  -11. 

2.    -8  take  6. 

6.   6  take  -  8. 

10. 

13  take  - 17. 

3.   8  take  -  6. 

7.   11  take  - 11. 

11. 

-12  take  7. 

4.-8  take  -  8. 

8.    - 11  take  +  11. 

12. 

0  take  64. 

Subtract  the  following  numbers  : 

13.    11  a^                      16.      a^hc^ 
11  a^          .                    2  a'bc" 

19. 

Had 
21  ad 

14.        12  x^y 
-12x^y 

17.        36  mnp 

—  22  mnp 

20. 

-^la^h^ 
+  41a262 

15.    —     a^?/ 
-2a^2/ 

18.          0 

-12  xyz 

21.  Subtract  17  xyz  from  6  xyz. 

22.  Subtract  the  sum  of  6  a;  and  7  x  from  — 12  x. 

23.  From  26  a  +  38  6  + 12  c  subtract  26  a  - 14  6  + 18  c,  and 
check  the  answer. 

24.  From  7  a^  + 12  a  +  3  subtract  2  a^  + 13  a  —  7,  and  check 
the  answer. 

25.  From  y?  —  lxy-\-y'^  take  y?~l  xy  —  y^. 

26.  From  -  81  a^ + 12  a&  -  7  V"  subtract  81  a' + 22  a6  -  14  h\ 


28  ELEMENT ABY  ALGEBRA 

27.  From  6a-7?>  +  8c-9d  take  12  a- 7  & -12c  +  17d 

28.  From  6  a:-3-17  x^  +  S  x-3,  subtract  3-7  aj  +  5  a^-9  a^. 

29.  From  7a;^-2a^  +  9  subtract  4-7  x  +  3  a^-9  a;^ 

30.  From  a«  -  3  a^6  +  3  ah^  -  h""  take  -  a^  +  3  a-6  -  3  a T^^  +  ft^ 

31.  From  ab —  'bc-\-cd  —  da,  subtract  ah -{- he -{- cd -\- da. 

32.  From  -2  +  8  aj3-4  ar-2  a;  take  2  ar^-7  a^4-3  a;. 

33.  From  m^  —  S  mn  +  6  n^  take  6  n^  —  7??,^  +  7  mn. 

34.  From  a;  +  ?/  ^^^^  x -{- y  -{- z -\- u. 

35.  From  19  a5  + 17  5c  — 14  ac  take  17  ah  + 19  5c  + 14  ac. 

36.  Subtract  x-\-y  from  a;  — 2/ +  2  —  '?^. 

37.  Subtract  7  a^-6  a^-5a;  +  2  from  7  -  6  x-S  a^  + 2  a^. 

38.  Subtract  2ax  —  7hy  —  5xy  from  2by  —  2  ax— 5  xy. 

39.  From  2  x  +  y  take  x-{-y  -\-z. 

40.  From  a;^  +  l  take  —  x"^  —  a?  —  x^  —  x. 

41.  From  —  a;'*  — a^^  — aj  subtract  aj^  — 1. 

42.  Subtract  a-\-h-{-  c  from  d. 

43.  From  a;  +  Va;  —  V^/  take  2  x  — 2Va;  — 2V^. 

44.  Subtract  Vaa?  +  V^x  —  Vca?  from  ^ax  +  ^hx  —  ^cx. 

45.  From  6(a;  +  2/)  +  4 Va;  —  Va;  +  ?/  subtract  7(a;  +  y)  —  5Va; 
+  7V^+^. 

46.  Subtract  a^  +  ia;^  — iaj  — |  from  2a;^— -i-aj^  +  ^a;  — |. 

47.  Subtract  ia^-f  a^-f  a-f  from  ^a^ +  ^a^  -  ^a  +  1. 

REVIEW  EXERCISE  I 

1.  To  the  sum  of  3  a  —  4  5  +  c  and  6a  —  26  —  7c  add  the 
sum  of  — 7  a  —  66  —  c  and  —  a  +  6  —  8  c. 

2.  From  the  sum  ofa  —  26  +  3c  and  2a  —  36  +  4c  subtract 
4a  —  56  +  6  c. 

3.  From  x-{-y  subtract  the  sum  of  2x  —  y-\-z  and  x  —  3y 
+  4  2. 


ADDITION,   SUBTRACTION,  AND  PARENTHESES     29 

4.  From  the  difference  of  a^—x  and  x—1  subtract  a^H-x+l. 

5.  Subtract  the  sum  of  5x^—6x  +  7  and  — 40^^  —  6a;  +  10 
from  x^  —  6x-\-16. 

6.  Subtract  the  difference  of  4  m^  —  2  m  —  7  and  3  m^  —  2  m 
+  7  from  unity. 

7.  Subtract  the  difference  of  6  a^  —  7  a  +  2  and  —  4  a^  4-  5  a 
4-  2  from  zero. 

8.  Subtract  the  sum  of  a  +  6  and  a  —  b  from  a-{-b-\-c. 

9.  Subtract  the  sum  of  x-j-y-\-z  and  x  —  y  —  z  from  the 
difference  of  x—y-^z  and  —x-\-y-\-z. 

10.  What  expression  must  be  added  to  6  a  +  2  5  to  produce 
7a-36? 

11.  What  expression  must  be  added  to  o?  —  a  to  produce 
a  +  6? 

12.  What  expression  must  be  subtracted  from  3  a  +  2  6  +  c 
to  produce  2  a  —  6  —  c  ? 

13.  From  what  expression  must  a  +  6  be  subtracted  to  pro- 
duce a  difference  of2a  —  36  +  c? 

14.  What  must  be  added  to  — 3a4-25  —  cto  produce  zero  ? 

15.  What  must  be  subtracted  from  a  —  &  to  produce  zero  ? 

16.  By  how  much  does  a-\-h  -\-c  exceed  a  —  6 -f- c ? 

17.  What  number  is  less  than  3a  +  26  by  —Sa  +  2b? 

18.  Add  3  X  2%  -5  X  2%  and  -9  x  2\ 

19.  Represent  the  sum  of  6  x  2^-  7  x  2^  +  2«  +  3  X  2^  as  a 
multiple  of  2^. 

20.  Add  135  X  3^-  140  x  3^  +  20  x  3^-24  x  3^ 

21.  A  is  a  years  old.     How  old  will  he  be  10  years  hence  ? 
a  years  hence  ? 

22.  A  is  10  years  old.     How  old  will  he  be  in  a  +  6  years  ? 
In  2  a  —  10  years  ? 

23.  A  was  n  years  old  when  B  was  born.     How  old  will  A 
be  when  B  is  2  n  years  old  ? 


30  ELEMENTARY  ALGEBRA 

If  a-\-b  +  c  =  m,  a  +  b  —  c  =  n,  2a  — 36  +  4c=|),  find 

24.  m  +  n.  2Q.    m -\- n -\- p.  28.    m-\-n—p. 

25.  m  —  n.  27.    m  —  n+p. 

29.  What  is  the  remainder  obtained  by  subtracting  any 
number  from  another  one  which  is  smaller  by  10? 

30.  If  6  a -{-2  b  is  the  subtrahend  and  2  a  — Sb  the  re- 
mainder, find  the  minuend. 

31.  Can  you  discover  a  short  method  for  adding  the  ten 
consecutive  numbers,  6  7  3  2  4  0,  673241,  673242,  and  so 
on  to  673249? 

Hint.     Consider  each  number  a  binomial. 

32.  If  3.2^-5.2^  +  7.2^  =  2^.0;,  findaj. 

SIGNS  OF  AGGREGATION 

42.  By  using  the  signs  of  aggregation,  additions  and  sub- 
tractions may  be  written  as  follows : 

a-\-(-\-b  —  c-{-d)  =  a-\-b  —  c  +  d. 
a—(-\-b  —  c-\-d)  =  a  —  b-\-c  —  d. 

Hence  it  is  obvious  that  parentheses  preceded  by  the  +  or 
the  —  sign  may  be  removed  or  inserted  according  to  the  fol- 
lowing principles : 

43.  I.  A  sign  of  aggregation  preceded  by  the  sign  -\-  may  be 
removed  or  inserted  without  changing  the  sign  of  any  term. 

II.  A  sign  of  aggregation  preceded  by  the  sign  —  may  be  re- 
moved or  inserted  provided  the  sign  of  every  term  inclosed  is 

changed. 

E.g.  a-\-(b  —  c)=a-\-b—  c. 

a4-(— 6  +  c)=a  — 6  +  c. 
a  —  (+6  —  c)=a  —  6-l-c. 
a b  —  c  =  a-\-b-{-c. 


ADDITION,   SUBTB ACTION,   AND  PARENTHESES     31 

44.  If  there  is  no  sign  before  the  first  term  within  a  paren- 
thesis, the  sign  +  is  understood. 

a  —  (6  —  c)  =  a  —  6  4-  c. 

45.  If  we  wish  to  remove  several  signs  of  aggregation,  one 
occurring  within  the  other,  we  may  begin  either  at  the  inner- 
most or  outermost.  The  beginner  will  find  it  most  convenient 
at  every  step  to  remove  only  those  parentheses  which  contain 
no  others. 


Ex.     Simplify  4a- ;(7a  +  56)-[- 66 +(-26-a-6)]! 


4a-{(7a4-5  6)-[-66  +  (-26-a-^)]} 
=  4a-{7a  +  66-[-6&  +  (-26-a  +  &)]} 
=  4a-{7a  +  56-[-66-26-a  +  6]} 
=  4a-{7a  +  56  +  66  +  26  +  a-6} 
=  4a-7a-56-66-26-a  +  6 
=  —  4  a  —  12  6.        Answer. 

EXERCISE  15 
Simplify  the  following  expressions : 

1.  a +  (6-2  e). 

2.  a -(6 -2a). 

3.  a2-(2  62  4-a2_c2). 

4.  a^  -  (+  a^  _  b^), 

5.  a-(—a-2b'^c)-^(-a  +  b). 


6.    2x-x-y-[-(—x  +  y). 


7.  4:X-\-  -\-Sx-i-y. 

8.  6a^-{-9Ax-7-(5x'-\-3x)  +  6. 

9.  (6a  +  66)-(5a-46)  +  (3a-26). 


10.  17 X  —  (16 X  -\-  y)  +  (4:X  —  y  -{-  z)  —  x  -{- z. 

11.  (a  +  &  +  c)  —  (a  +  6  —  c)  —  (a  —  &  +  c). 


32  ELEMENTARY  ALGEBRA 

12.  a +  [&-(«-&)]. 

13.  a  +  h-l{b  +  d)-{a-'b)']. 


14.  m— (ti— p)4-[3m  — 3?i  — 6m]. 

15.  a  — [a  — Ja  — (— a)J]. 

16.  x  —  \_^y  +  {^z  —  (z  —  x)-\-y]  —  2x]. 

17.  —  [m  — (m4-7i)  — (m  — Ti)  — (— m  + ?^i)]. 

18.  12a-[(a-\-h)-\h-{a-h)']-a\. 

19.  2x  — faj  — (x  — 2/)  — [cc  — £c  — 2/]— 2/1- 


20.    12-2a-J-a-[2a-(a-7-a)]|. 


21 .  14  -  3  a  -  J  9  a  -  [10  a  -  (11  a  -  6  -  6  a)  J ;. 

22.  a-[-J-(-a)j]. 


23.    a-3-[-5-(-a  +  a  +  6)|]. 


24.  a?  +  2/-[-(a^-2/)+S-^+(^-^-2/)n- 

25.  l-J-a-(a  +  l)-[-a-(a-a-l)]|. 

26.  a-(-{-[-(-a)]S). 


27.  l-(-[a+(-a  +  l)J)-Jot-c*-l5- 

28.  6m  +  J4m-[8n-(2m  +  4n)-22n]-7wJ 

+  [9  m  -  (3  71 4-  4  m)  + 14  w], 

29.  1247  -  [1722  -  J 1722  +(933  - 1247)  |]. 


30.  From  a-\-\{4.  —  'b)  +  (a  —  ^)—a  —  l\  subtract 

a_J(6-&)  +  (6a-6)-(5a-7)J. 

31.  From  the  sum  of  a +  Ja  — (6  — c)  5  and 

—  a  +  [4  a  —  (5  &  +  c)]  subtract  a  —  (b  —  c). 


32.  Simplify  4a  —  [6  6+(3a  —  c)  —  f56  —  c  —  aj]  and  check 
the  answer  by  substituting  a  =  3,  6  =  2,  c  =  1  in  the  question 
and  the  answer. 


33.    Simplify     9a- [— 7a +  {5  6  — (a  — 6)  +  «— &1]     and 
check  the  answer  by  the  substitution  a  =  1,  6  =  2. 


ADDITION,   SUBTRACTION,  AND  PARENTHESES     33 

46.   Signs  of  aggregation  may  be  inserted  according  to  §  43. 

Ex.  1.    In  the  following  expression  inclose  the  second  and 
third  and  the  fourth  and  fifth  terms  respectively  in  parentheses : 

a  —  b  -\-  c  +  2d  —  e 
=  a-(h  -c)  +  (2£?-  e). 

Ex.  2.    Inclose  in  a  parenthesis  preceded  by  the  sign  —  the 
last  three  terms  of        2a  +  6-5c  +  2(2 

=  2a-(-6  +  5c-2<i). 

EXERCISE  16 

In  each  of  the  following  expressions  inclose  the  last  three 
terms  in  a  parenthesis  : 

1.  e  +  6  +  c  — d  3.   2a;  — 6a^  — 7a^H-5a;l 

2.  x-2y  +  3z-Ad.  4.   2p-q-\-p^-q\ 

In  each  of  the  following  expressions  inclose  the  last  three 
terms  in  a  parenthesis  preceded  by  the  minus  sign : 

5.   a-b-c-d.  S.   a-\-o?  —  h  +  h\ 

Q.    a~h  +  c  +  d.  9.   1  +  a  — 26  — c  — d 

7.   a  +  26  — 5c  — cZ.  10.   a;  — ?/4-2!  +  m  +  n+p. 

EXERCISES  m  ALGEBRAIC   EXPRESSION 

EXERCISE  17 

Write  the  following  expressions : 

1.  The  sum  of  the  squares  of  a  and  6. 

2.  The  square  of  the  sum  of  a  and  h. 

3.  The  difference  of  the  cubes  of  x  and  y. 

Note.     The  minuend  is  always  the  first,  and  the  subtrahend  the  second, 
of  the  two  numbers  mentioned. 


84  ELEMENTABY  ALGEBRA 

4.  The  difference  of  the  cubes  of  y  and  x, 

6.  The  cube  of  the  difference  of  x  and  y. 

6.  The  product  of  a  and  h. 

7.  The  product  of  the  cubes  of  a  and  h. 

8.  The  cube  of  the  product  of  a  and  6. 

9.  The  product  of  the  sum  and  the  difference  of  a  and  />. 

10.  Six  times  the  square  of  the  difference  of  a  and  h  dimin- 
ished by  the  quantity  a  minus  h. 

11.  The  product  of  the  difference  of  a  and  h  and  the  quan- 
tity a^  -  2  6^  +  c^. 

12.  The  sum  of  a  and  h  diminished  by  the  difference  of  a 
and  h. 

13.  a  cube  minus  the  quantity  2a^  minus  6?/^  plus  1  (?  plus 
the  quantity  —x-\-y. 

14.  The  sum  of  the  cubes  of  a,  —  b,  and  c  divided  by  the 
difference  of  a  and  c. 

Write  algebraically  the  following  statements : 

15.  The  sum  of  a  and  b  multiplied  by  the  difference  of  a  and 
b  is  equal  to  the  difference  of  a^  and  6^. 

16.  The  difference  of  the  cubes  of  a  and  b  divided  by  the 
difference  of  a  and  b  is  equal  to  the  square  of  a  plus  the  prod- 
uct of  a  and  b,  plus  the  square  of  b. 

17.  The  difference  of  the  squares  of  two  numbers  divided  by 
the  difference  of  the  numbers  is  equal  to  the  sum  of  the  two 
numbers.     (Let  a  and  b  represent  the  numbers.) 


CHAPTER   III 

MULTIPLICATION 

MULTIPLICATION  OF  ALGEBRAIC  NUMBERS 

EXERCISE  18 

1.  If  a  man  makes  $  15  a  day,  how  many  dollars  will  he 
make  in  5  days  ? 

2.  If  a  man  loses  f  15  a  day,  how  many  dollars  will  he 
make  in  terms  of  algebra  in  5  days  ?  (Denote  gain  by  +,  and 
loss  by  — .) 

3.  If  from  a  man's  fortune  $  15  are  deducted  5  times,  how 
much  in  terms  of  algebra  does  he  make  ? 

4.  If  from  a  man's  debts  $15  are  deducted  5  times,  how 
much  does  he  gain  ? 

5.  Express  each  of  the  Exs.  1-4  as  a  multiplication  example, 
considering  gain  as  positive,  and  loss  or  debts  as  negative. 

6.  If  we  denote  three  days  hence  by  -f-  3,  by  what  must  we 
denote  three  days  ago  ? 

7.  If  we  denote  northerly  motion  as  positive,  and  three 
days  hence  as  +  3,  express  the  following  as  multiplication 
examples  with  algebraic  symbols : 

(a)  A  ship  sailing  north  at  the  rate  of  3°  per  day  and  crossing 
the  equator  to-day  will,  in  6  days,  be  18°  north  of  the  equator. 
(6)  The  same  ship  6  days  ago  was  18°  south  of  the  equator. 

(c)  A  ship  sailing  south  at  the  rate  of  3°  per  day  and  crossing 
the  equator  to-day  will,  in  6  days,  be  18°  south  of  the  equator. 

(d)  The  same  ship  6  days  ago  was  18°  north  of  the  equator. 

86 


6b  ELEMENTARY  ALGEBRA 

8.  If  the  signs  obtained  for  the  products  in  the  preceding 
examples  were  generally  correct,  what  would  be  the  value  of 

6x3,  6x(-3),  (-6)x3,  (-6)x(-3)? 

9.  State  a  rule  by  which  the  sign  of  the  product  of  two 
numbers  can  be  obtained. 


47.  Multiplication  by  a  positive  integer  is  a  repeated  addition ; 
thus,  4  multiplied  by  3,  or  4  x  3  =  4  +  4  +  4  =  12, 

-4*multiplied  by  3,  or  (_4)  x  3  =  (-4)  +  (-4)  +  (-4)=-12. 

The  preceding  definition,  however,  becomes  meaningless  if 
the  multiplier  is  a  fractional  or  negative  number.  To  take  a 
number  2|  or  —  7  times  is  just  as  meaningless  as  to  go  to  bed 
2|  times  or  to  fire  a  gun  —7 times. 

A  more  useful  definition  of  multiplication  which  may  be  used 
in  nearly  all  multiplication  problems,  is  the  following : 

48.  Multiplication  is  the  operation  of  finding  a  number  that 
has  the  same  relation  to  one  factor  (multiplicand)  as  the  other 
factor  (multiplier)  has  to  1. 

Thus  f  is  obtained  from  1,  by  taking  one  fifth  of  unity  three 

*i^es,  or  8  _  1   .  1   .  1 

3"  —  o  ^^  5"  +  :?• 

Therefore  6  multiplied  by  f  is  obtained  by  taking  one  fifth 
of  6  three  times,  or       ^      ,_,.,.  , 

49.  The  product  of  4  multiplied  by  —  3  is  obtained  from  4 
in  the  same  manner  in  which  —  3  is  obtained  from  1. 

But  —  3  =  —  1  —  1  —  1,  i.e.  —  3  is  obtained  by  subtracting 
three  times  1. 

Therefore,  (+4)x(-3)  =  -(+4)-(+4)-(4-4)  =  -12. 
and  (-4)x(-3)=-(-4)-(-4)-(-4)=  +  12. 


MULTIPLICATION  .      37 

In  multiplying  algebraic  numbers  we  have  therefore  the 
following  cases  :  4  X  3  =12 

(-4)x3  =  -12.     (§47.) 
4x(-3)^-12. 
(_4)x(-3)  =  +  12. 

50.  Evidently  the  particular  numbers  chosen  do  not  affect 
the  sign  of  the  answer,  and  hence  the  preceding  illustrations 
lead  to  the 

Law  of  Signs :  Tlie  product  of  two  numbers  with  like  signs  is 
positive  ;  the  product  of  two  numbers  with  unlike  signs  is  negative. 

To  obtain  the  absolute  value  of  a  product  we  multiply  the 
absolute  values  of  the  factors. 

Thus  (+  «)(+•&)  =  +  (a&), 

(+a)(-&)  =  ^(a&). 
(-«)(+ 6)  =  -(a?>).    • 
{-a){-b)  =  -\-{ab). 

51.  Some  fundamental  laws  which  are  correct  for  arithmetic, 
we  shall  assume  for  algebra. 

I.  Commutative  Law. 

(a)   For  addition,         a  +  b  —  b-\-a, 
e.g.     4  +  5  =  5  +  4. 
(6)   For  multiplication,    ab  =  ba, 
e.g.     4.5  =  5-4. 

I.e.  Algebraic  numbers  may  be  added  (or  multiplied)  in  any 
order. 

II.  Associative  Law. 

(a)   For  addition, 

a  +  6  +  c  =  a  +  (6  +  c)  =  (a  +  &)  +  c, 
e.g.     4  +  5  +  6  =  4  +  11  =  9  +  6. 


38  ELEMENTABY  ALGEBRA 

(b)   For  multiplication, 

abc  =  a(bc)  =  (ac)b  =  (ab)c. 
e.g.    2.3.4  =  2.12  =  8.3  =  6.4. 

I.e.   The  sum  (or  the  product)  of  three  or  more  algebraic 
terms  does  not  depend  upon  the  grouping  of  the  terms. 

III.   For  the  Distributive  Law  see  §  55. 

These  laws  may  sometimes  be  used  to  shorten  the  arithmetical  work, 
e.g.  14  .  (-  11)  .  Q)  .  (-  j\)  =  (14  .  1)  .  (-  11  .  -  ,2^)  z.  2  X  2  =  4. 

EXERCISE  19 

Find  the  values  of  the  following  products : 

1.  6x(-3).  4.    (-15)x(-5). 

2.  (-7)x(  +  7).  5.    4x(-13). 

3.  (-22)  X  (+7).  -  6.    13  X  (-3). 

Note.    If  no  misunderstanding  is  possible  the  parentheses  about  fac- 
tors are  frequently  omitted. 

7.  -16X-4,  13.    (-2)3. 

8.  +4.-4.  14.    (-3)V 

9.  -7.(21.).  15.    (_i)w 

10.  21. -|.  16.    01''. 

11.  -5.27.20.  17.    (-10)«. 

12.  (-ioi).-f.-9.  +  i.     18.  (-^y. 

19.  Formulate  a  law  of  signs  for  a  product  containing  an 
even  number  of  negative  factors. 

20.  Formulate  a  law  of  signs  for  a  product  containing  an 
odd  number  of  negative  factors. 

If  a  =  3,  6  =  —  2,  c  =  1,  d  =  0,  and  x  =  5,  find  the  numerical 
value  of : 

21.  6  abc.  23.    7  ab^cK 

22.  Aa^bK  24.    -3aV. 


MUL  TIPLICA  TION  39 

25.  -Id^dx.  32.  4a2-562-6c. 

26.  21a6V\  33.  4a6-55H6cd 

27.  -(2a6c)2.  34.  {a  +  hY-(^. 

28.  -2(a6c)2.  35.  a  + J  6-- (c  +  d)^}. 

29.  +2a(hcf.  36.  7a6V- 9  ftc^  +  Tc. 

30.  -4  6^  37.  V3a4- V-2  6+ V^. 

31.  (-4  c)*.  38.  2^  —  0?, 

MULTIPLICATION  OF  MONOMIALS 

52.  By  definition,  a?  =  a  •  a  -  a,  and  a^  =  a  -  a  »  a  -  a  •  a. 
Hence  a^xa^  =  a'a'axa-a'a'a-a  =  a^,  i.e.  a^\  Or  in 
general,  if  m  and  w  are  two  positive  integers, 

a"*  X  a**  =  (a  •  a  •  a  •••  to  m  factors)  •  (a  •  a  •  a  •••  to  n  factors) 
=  a  a  •-•  to  (m-\-n)  factors. 

This  is  known  as 

The  Exponent  Law  of  multiplication :  TJie  exponent  of  the  prod- 
uct of  several  powers  of  the  same  base  is  equal  to  the  sum  of  the 
exponents  of  the  factors. 

53.  To  multiply  two  monomials,  5  a^b^c  and  —  7  a^b*d\  we  apply 
the  laws  of  association  and  commutation, 

5a^b^c  X  -Ta^b'd'  =  (5  •  -7)  •  (a'  •  a')  -  (b^  -  b')  -  c  -  d* 
=  - 35  a'b'cd^. 

EXERCISE  20 

Express  each  of  the  following  products  as  a  power : 

1.  a^-a^.  4.    (x-{-yf-(x  +  yy, 

2.  b''(-b').  5.    23.2^. 

3.  (xyy-(xy)\  6.    143^^.143^=*. 


40  ELEMENT ABY  ALGEBRA 


8.    x'^^^'X"-^.  11.    a 


n-l 


a. 


9.    22.23.2^.2.  12.    a«-2.a.(-a). 

Perform  the  multiplications  indicated : 

13.  Zx'^xy.  19,  IxyZ'Smn. 

14.  Zxy  '  {-5xy^.  20.  25  x-y-z^  -  {-S  xyz). 

15.  -TaS-Sa^ftV.  21.  (- 6  icy^2^) .  (_3  ^^^4^^^ 

16.  22  c«263 .  (_  5  ab^c).  22.  6  ao^V  .  (-  7  byh). 

17.  (-lla46V).(-27a^6^c).  23.  7  a^^^^  •  (9  «6V). 

18.  (-  4  tt^6c)  .  2  a'^62(;.^.  24.  (-  12 ^gr)  .  12  mnr. 

25.  3  a'b'c' .  (-  4  a6c)  •  (-  5  aV). 

26.  (- 2  a'bx)  '  (- 2  ab'x)  '  2  aba^. 

27.  7  mn  •  (—  5  np)  •  (—  2pq). 

28.  (-3a3).(-2  63).(_c). 

29.  2(x  +  2/)'-3(.'c  +  2/)-3(^  +  2/). 

30.  3(a  -by-  -  2(a  -  by  •  (a  -  6)1 

31.  6a\7n  +  7i)  •  (-7  ab). 

32.  (-  2  a^ft'")  .  (a^ft"*).  34.    2«  •  22«  .  2^''  -  2\ 

33.  a"+i .  a«-2  -a.  35.    («  +  6  +  c)  •  (a  +  6  +  c)". 

MULTIPLICATION  OF  A  POLYNOMIAL  BY  A  MONOMIAL 

54.  If  we  had  to  multiply  2  yards  and  3  inches  by  3,  the 
results  would  evidently  be  6  yards  and  9  inches.  Similarly  the 
quadruple  of  a  +  2  6  would  be  4  a  +  8  &,  for 

4(a  +  2  6)  =  (a  +  2  6)  +  (a  +  2  5)  +  («  +  2  &)  +  (a  +  2  5) 
=  4  a  4-  4(2  6)  =  4  a  +  8  6. 


MULTIPLICATION  41 

55.  This  principle,  called  the  distributive  law,  is  evidently 
correct  for  any  positive  integral  multiplier,  but  we  shall  as- 
sume it  for  any  number. 

Thus  we  have  in  general 

a(b  +  c)  =ab-^ac. 

56.  To  multiply  a  polynomial  by  a  monomial,  multiply  each  term 
by  the  monomial. 

-  3  a%{Q  a^hc  +  2  6c  -  1)  =  -  18  a'^h'^c  -  6  a%'^c  +  3  a%. 
EXERCISE  21 

Perform  the  multiplications  indicated. 

1.  3a;(ic  +  2/  +  2).  6.    2 xyzij x'y  +  2 yz" -  Z x'z). 

2.  -^xy{a?-y^-l).  7.    -^xz{^x'z-Zxz'^-{-2xz). 

3.  2mn{m''-n').  8.    -2hc{a^ -h^-^c^ +  d^). 

4.  4a26V(-4a+2a6c-c^).      9.    -5xhj(l xyz-5xyh''-2z). 

5.  -^a\-bd'-ba-b).   10.    Q  xy\2  x'' -  2  xy  -  y^  -  1). 

11.  4.p^q{-2pq-{-2p'-3q^  +  l). 

12.  f  a2(4  a^  -  2  a6  -  4  6^). 

13.  -a^l-tt^  +  tt^'^-a^). 

14.  6"*(1  +  6  -  6«  +  h"^), 

15.  (a  +  6)T(a  +  hf  +  (a  +  &)'  +  («+&)  +  1]. 

16.  By  what  expression  must  x  be  multiplied  to  give  the 
product  ax  —  hx -{-ex? 

17.  Express  3  aaj  +  4  6a;  —  ex  as  a  product. 

18.  How  many  x  are  contained  in  mx  —  nx? 

19.  Express  the  sum  of  +  n^x  and  —  m^x  as  a  multiple  of  x. 

20.  Find  the  factors  of  a6  +  ac  —  3  a. 

21.  Find  the  factors  of  3  ab-\- S  be —  3  ac. 

22.  Find  the  factors  oi  3x^  —  6x^  —  15x. 

23.  Find  the  factors  of  5  a^b  -  15  a%^  -  60  a6c. 


42  ELEMENTARY  ALGEBRA 


MULTIPLICATION  OF  POLYNOMIALS 

57.  Any   polynomial   may   be  written  as  a   monomial   by 

inclosing  it  within  a  parenthesis.     Thus  to  multiply  a—  bhy 

x-\-y—z,  we  write  (a—  b)(x r{- y  —  z)  and  apply  the  distributive 

law. 

(a  —  b)(x-\-y  —  z)=  x(a  —  6)  +  y(a  —b)  —  z{a  —  b) 

=  {ax  —  bx)  +  (ay  —  by)  —  (az  —  bz) 

=  ax—bx-\-ay  —  by  —  az  -{-  bz. 

58.  To  multiply  two  polynomials,  multiply  each  term  of  one  by 
each  term  of  the  other  and  add  the  partial  products  thus  formed. 

The  most  convenient  way  of  adding  the  partial  products  is 
to  place  similar  terms  in  columns,  as  illustrated  in  the  follow- 
ing example : 

Ex.  1.   Multiply  2a-Sbhja-5b. 

2a-Sb 
a  —  5b 


2a^-Sab 

2  a-2  -  18  ab  +  15  b^        Product. 

59.  If  the  polynomials  to  be  multiplied  contain  several 
powers  of  the  same  letter,  the  work  becomes  simpler  and  more 
symmetrical  by  arranging  these  expressions  according  to  either 
ascending  or  descending  powers. 

Ex.2.   Multiply  2  + a^-a-3a2  by  2a-a^  +  l. 

Arranging  according  to  ascending  powers : 

Check.    If  a  =  1 
2-a-Sa^  +  a^  =-1 

l  +  2a-a^  =     2 

2  -     a-Sa^  +  a^ 
+  4«-2a2_6a3  +  2a* 

-  2  gg  +  q3  +  3  a4  _  a5  

2  +  3a-7a2-4a3  4.5a4_(^6  __2 


MULTIPLICATION  43 

60.  Examples  in  multiplication  can  be  checked  by  numerical 
substitution,  1  being  the  most  convenient  value  to  be  substi- 
tuted for  all  letters.  Since  all  powers  of  1  are  1,  this  method 
tests  only  the  valTies  of  the  coefficients  and  not  the  values  of 
the  exponents.  Since  errors,  however,  are  far  more  likely  to 
occur  in  the  coefficients  than  anj^where  else,  the  student  should 
apply  this  test  to  every  example. 

Ex.  3.   Multiply  3a;y4-6aj^2/H-/— 2a;2/^  by— 2  xf-\-^x'^—6y\ 
Arranging  according  "to  descending  powers  of  x  : 

Check.    If  oj  =  1,  y  =  1 


6x^y-\-Sx'^y^-2xy^-\-y^ 

=  8 

4  x4  -  2  x?/3  -  5  2/* 

=  -3 

24  x^y  +  12  x^y^  -  8  x^y^  +  4  x^y^ 

-12xV 

-6xY+   4:X^y^-2xy^ 

-SOx^y^ 

-ISxV  +  lOx^/^-Syio 

2^x^y                -  38x52/5 +  4  xV  ■ 

-6xV-lla:¥+    8x^9-52/10 

=  -24 

61.  The  degree  of  a  term  is  equal  to  the  number  of  literal 
factors  contained  in  the  term.  Hence  the  degree  of  a  term  is 
also  equal  to  the  sum  of  the  exponents  of  the  literal  factors. 

—  17  a^'^cd  is  a  term  of  the  7th  degree,  since  the  sum  of  the  exponents 
is  3  +  2  +  1  +  1. 

62.  A  homogeneous  polynomial  is  a  polynomial  whose  terms 
are  all  of  the  same  degree  ;  as  6  nfyz  —  3  xyh^  —  2z^. 

The  product  of  two  homogeneous  polynomials  must  be  homo- 
geneous, since  the  degree  of  every  term  in  the  product  must  be 
equal  to  the  sum  of  the  degrees  of  the  factors.  This  fact  may 
be  used  as  an  additional  check  in  some  examples. 

EXERCISE   22 

Perform  the  following  multiplications  and  check  the  results: 

1.  (2a-5&)(7a  +  46).  4.    (5x-3y)(6x- 5y). 

2.  (9rH-7s)(2r-5s).  5.    (2  a-3  6)(4  a-5  6). 

3.  (Sp-3q){5p-^7q).  6.    {7  x-5  y)(3  x  +  4.y). 


44  ELEMENTARY  ALGEBRA 

7.  {Qx--ly)(2x~^y).  12.    (8  r -3  s)(2  r  +  5  s). 

8.  {a-Q>h){a-7h).  13.    {^  x- 4.y){^  x~l  y). 

9.  (6a-9)(6a  +  9).  14.    (6  o^i/ +  7,;2)(-3  o^t/- 4  ^). 

10.  {^u-\-2v){^u-lv).  15.    (-aaj-2/)(-aa;H-2/). 

11.  (9i)-4g)(2p-3g).  16.    (a6c  +  5)(6a5c  -  7). 

17.  (9  ahc  -  4  a262c2)  (9  abc  +  4  a^^ V). 

18.  {x'-\-x  +  l){x-l). 

19.  (-3a  +  5a2  +  2)(-5a-4). 

20.  (3a^2_4_5^^^^2-3a^). 

21.  (7-9r  +  3r2)(5-8r). 

22.  (7r2-8r  +  5)(7-9r). 

23.  {na'-3ah+2-b''){^a?-bah-4.h\ 

24.  (4aj''^+l-ic)(ic2_3a;4_9). 

25.  (9  aj2  -  7  a^?/  +  8  y'') {9'x'  -Sy^  +  1  xy). 

26.  {Qu'^  +  ^'i^-^uv){Qu' -{-BUV-4.V'). 

27.  (a4-a3  +  l4-a')(a-l). 

28.  (a2-2a5  +  52)(^2_|_2«,5  +  52)^     . 

29.  (3a-4&-5c)(4a  +  5  6-3c). 

30.  (5a;-72/4-9;2)(5a;  +  72/-9;j^). 

31.  (6r  +  5s-4f)(6r-5s  +  4^). 

32.  (7a-6-2c)(-7a  +  6-2c). 

33.  {x'-Sx^-4.x'  +  6x  +  ^)(x-l). 

34.  (a-5a3^-3<x2_^,l)(c^2_^;J^_l_^^J_ 

35.  (a»-2a4_i)(a_a5^1). 

36.  (a*  -  4  a-'^ft  +  6  a252  _  4  ^53  _^  ^4^(^2  _2ab+  b^. 

37.  (-«:»  + a^2/'- A' 4-.v')(«^/  +  a^' 4-2/0- 

38.  (x'+x^-^x-{-l)(cf^-3x-l). 

39.  (a^-i^^-a,4  +  2/)(a^3_^^^)^ 

40.  (a;^  +  a;«  +  a^  +  a;^  +  .T3  f  x'2  +  a;  +  l)(ic-l). 


MULTIPLICATION  46 

41.    (a^  +  ^a-f |)(2a  +  i).  47.    (a^+''  +  2  6)(a^+*-2  6). 

43.  (a"  +  l)(a"-l).  49.    (a -2  6)1 

44.  {x^-^y'^)(x^  —  y'^).  50.    (tt  +  6  —  2  c)2. 

46.    (.T"-i-l)(aJ-l).    ^  51.    (a-26  +  c-3d)«. 

46.    (a;2"4-a;'»2/"'4-?/2'»)(a;»--i/'»).     52.    (a- -l)(a  +  l)(a  -  1). 

53.  (a;  +  ?/)(a;-2/)(a.'-^)(a?  +  2/). 

54.  (a-b){p?-2ah-^h'')(a  +  h). 

55.  (a^  +  2/2)(a;2_  2/2^^(^,4  _^4), 

56.  (a  +  l)(a  --  l)(a2  +  l)(a^  +  !)(«'  +  1). 

57.  {m  ~n){m  -{-  n)(m^  +  mn  +  inF){m^  —  mn  +  ri^. 

58.  (/?  +  q){p'  -\-p'q  +pY  +  pq'^  +  g^)(p  -  q). 

59.  ((X-6)3.  61.    (m-ny. 

60.  (a;  +  2  2/)^  62.    (m  +  2w  +  3py. 

63.  (i.a2  +  ^a-l)(2a-4)(6a-2). 

If   J/'=  a  +  ^  +  c,    N=a-\-b  —  c,    P=a  —  h-\-c,    and    Q  = 
—  a  4-  6  4-  c,  find  the  following  product : 

64.  MN.  65.    PQ.  66.    NPQ. 
(For  method  of  detached  coefficients  see  Appendix  I.) 

SPECIAL  CASES  IN  MULTIPLICATION^ 

63.  The  product  of  two  binomials  which  have  a  common  term. 

(a;  +  2)(x  +  4)  =  iB2  +  2a;  +  4aj  +  8  =  a^  +  6a;  +  8. 

(a;  _  2)  (a;  +  4)  =  a^  -  2  a;  +  4  a;  -  8  =  aj2  _^  2  a;  -  8. 

(a;-2)(a;-4)  =  ar^-2a;-4a;  +  8  =  a^-6a;-f8. 

(4  a  +  7  6)(4  a  -  5  6)  =  16  a^  +  28  a6  -  20  a6-  35  h\ 

=  16a2  +  8a6-35  62. 


46  ELEMENTARY  ALGEBRA 

64.  TJie  product  of  two  binomials  which  have  a  common 
term  is  equal  to  the  square  of  the  comrnon  term,  plus  the  sum  of 
the  two  unequal  terms  multiplied  hy  the  common  term,  p)lus  the 
product  of  the  two  uriequal  terms. 

(5a  —  6  6)(5  a  —  9  &)  is  equal  to  the  square  of  the  common  term, 
25  a^,  plus  the  sum  of  the  unequal  terms  multiplied  by  the  common 
term,  i.e.  (— 15&)(5a)  =  — 75  a&,  plus  the  product  of  the  two  unequal 
terms,  i.e.  +  54  6^.    Hence  the  product  equals  25  a^  —  75  a&  +  54  b^. 

EXERCISE  23 

Multiply  by  inspection : 

1.  (x  +  3)(x-{-5),  16.  (x-}-y)(x-y). 

2.  (a-4)(a-3).  17.  (a' -^  5)(a' -  6). 

3.  (a-6)(a-4).  18.  (a^ -ob)(a^  -  eb). 

4.  (a-6)(a  +  4).  19.  (a^b' +  9)(a'b^-7). 

5.  (a-l)(a-2).  20.  (a'b'' -2c'')(a^b^ +  2c^. 

6.  (a-8)(a-4).  21.  (a'-7  b^a""-!  b'), 

7.  (aj-12)(a;  +  13).  22.  (1 +  2a;)(l +  3a;). 

8.  (a;-12)(a;-13).  23.  (a^ -}-2b^)(a^ -2b^). 

9.  (x  +  3)(x-\-3).  24.  (100  4-3)(100  +  1). 

10.  (x-25)(x-^).  25.  (1000-6)(1000  +  7). 

11.  (a;  +  25)(a;-4).  .  26.  103x105. 

12.  (x-^2y){x  +  3y).  27.  107x109. 

13.  (x-4:y)(x-2y).  28.  1008x1009. 

14.  (a-5  6)(a  +  4  6).  29.  999x1016. 

15.  (a-8)(a^8). 

30.  47,796  X  28,534  =  1,363,811,064.  Find  the  product  of 
the  two  next  higher  numbers.     (47,797  and  28,535.) 

31.  A  garden  285  yards  long  and  215  yards  wide  is  length- 
ened and  widened  by  one  yard.  Without  multiplying  285  and 
215,  find  the  increase  in  area. 


MULTIPLICATION  47 

32.  (2a  +  7)(2a-3). 

33.  (4aH-9)(4a-8). 

34.  (3a-4)(3a4-6). 

35.  (-a4-4)(-a-4). 

36.  (-a^  +  4a:)(-a^-7a;). 

37.  {2-lx)(2-Qx). 

38.  (a;-4)(5  +  a;). 

39.  (a;-2  6)(3  6  +  a;). 

47.  Find  two  binomials  whose  product  equals  x^  —  bx-\-Q. 

48.  Find  two  factors  of  a;^  -f-  9  a;  +  20. 

49.  Find  two  factors  oi  x^ —  llx-{-l^. 


40. 

(6  +  0^(^^-7). 

41. 

(a,_l)(a,  +  l)(a^_3). 

42. 

(^  +  2/  +  7)(a;  +  2/-3). 

43. 

(a,  +  2/-4)(a;4-2/-3). 

44. 

(a  +  &)l 

45. 

{a-hf. 

46. 

(a  +  b)(a-b). 

65.  Some  special  cases  of  the  preceding  type  of  examples 
deserve  special  mention : 

I.  (a  +  by  =  a'-\-2ab-\-b'. 

II.  (a-by  =  a'-2ab-{-b'. 

III.  (a  +  b)(a-b)  =  a'-b\ 
Expressed  in  general  language : 

I.  TJie  square  of  the  sum  of  two  numbers  is  equal  to  the  square 
of  the  first,  plus  twice  the  product  of  the  first  and  the  second,  plus 
the  square  of  the  second. 

II.  The  square  of  the  difference  of  two  number's  is  equal  to  the 
square  of  the  first,  mirius  twice  the  product  of  the  first  and  the 
second,  plus  the  square  of  the  second. 

III.  Tlie  product  of  the  sum  and  the  difference  of  two  numbers 
is  equal  to  the  difference  of  their  squares. 

The  student  should  note  that  the  second  type  (II)  is  only  a 
special  case  of  the  first  (I). 

Ex.  (4  a;8+ 7  y^y  is  equal  to  the  square  of  the  first,' i.e.  16  o^,  plus  twice 
the  product  of  the  first  and  the  second,  i.e.  bQx^y^.,  plus  the  square  of  the 
second,  i.e.  49  y^.    Hence  the  required  square  equals  16  a^  +  56  x^y^ + 49  y*. 


48  ELEMENTARY  ALGEBRA 


EXERCISE 

24 

Multiply  by  inspect] 

Lon : 

1. 

(x-\-yf. 

22. 

{x--y-y. 

2. 

(a +  4)1 

23. 

(9n  +  i)l 

3. 

{p-q)\ 

24. 

{x-^y){x-y). 

4. 

(n-^4)l 

25. 

(a2  +  4)(a2-4). 

5. 

{s-iy. 

26. 

{pq  +  q){pq-q). 

6. 

(a  +  1/. 

27. 

{ns''-f)(ps'-^i:^. 

7. 

(m2  +  2)2. 

28. 

(6mV  +  4)(6m3w3-4). 

8. 

{d^-lf. 

29. 

{xY-^z){x^f^4.z). 

9. 

{a  +  2hf.      . 

30. 

(4  a^?/V  -#)(4  a?yh^  +y), 

10. 

{x-7yf. 

31. 

(x-+r)(^"-r)- 

n. 

{d?^4.cy. 

32, 

(a!^--fll)(a^--ll). 

12. 

{m^-12ny. 

• 

33. 

(x^^7f){:^^7y^). 

13. 

(2x-7yy. 

34. 

(7n-2^2n4)(2n^  +  m2). 

14. 

C2a  +  9by. 

36. 

{a^^2ab)(2ab  +  a^). 

15. 

(4m-3n)l 

36. 

(d'^7)(~-7  +  d'). 

16. 

(2a^-3fy. 

37. 

(u''-7)(7-^u^). 

17. 

{ia'b'-iy. 

38. 

(cc?2^12/)(-12/+cd^). 

18. 

(4dV  +  3/0l 

39, 

l(a-^b)^c-]l(a  +  b)  +  cl 

19. 

(6xY-Sy'zy. 

40. 

[m4-n-p][w  +  n+2)]. 

20. 

(Jmnp^q^SnYq^y. 

41. 

(a-2b  +  Sc)(a~2b-8c). 

21. 

(m^-^iy. 

42. 

(40  +  1)1 

43. 

(100  -  sy. 

48. 

203^. 

53.    (40+2) (40-2). 

44. 

21\ 

49. 

9971 

54.   41x39. 

45. 

1021 

50. 

532 

55.    57  X  63. 

46. 

511 

51. 

m 

56.    997x1003. 

47. 

99^ 

52. 

96^ 

57.  6.5  X  7.5. 

58.  Prove  that  (n  -\- ff  =  n(n  i- 1)  +  -^. 

59.  Illustrate  Ex.  58  by  numerical  examples. 


MULTIPLICATION  49 

66.   The  product  of  two  binomials  whose  corresponding  terms 
are  similar. 

hj  actual  multiplication,  we  have 

5X    —       4:2/ 

IBx'  +  lOxy 

-12a?y-8/ 


15x^-   2a:?/-8/ 

The  middle  term  of  the  result  is  obtained  by  adding  the 
product  of  5x  '  2y  and  —4:y  -  3x.  These  products  are  fre- 
quently called  the  cross  products. 

Sx  +  2y 


>x  —  4:y 


10  xy  — 12  xy 
Hence  in  general,  the  product  of  two  binomials  whose  corre- 
sponding terms  are  similar  is  equal  to  the  product  of  the  first  two 
terms,  plus  the  sum  of  the  cross  products,  plus  the  product  of  the 
last  terms. 

EXERCISE  25 
Multiply  by  inspection : 

1.  (2:»  +  3)(x-5).    .  7.    (a2-7  62)(2a2-3&2). 

2.  (3x-2){x-l).  8.    (4a263-2)(3a26«  +  5). 

3.  (3^-2)(3a;-3).  9.    (2x'-3f)i2x'-5if). 

4.  (fix-l){2x^+Z).  10.    (2x'y-a){2x^y-la). 

5.  (5a-6)(a-2&).  11.    {-y?-\-2ah){2x'-3ah) 

6.  (lmn-\-p){2mn  —  3p).  12.    {px  —  ly){^y  —  lx). 

67.   The  square  of  a  polynomial. 

(a  Jrh  +  cf=  a"  -{-y  +  (?  -{-2  ah  ^2  ac  +  2hc. 

{a-h^c-df 

=  a^^b'-+c'^+d^-2ab+2ac-2ad-2bc+2bd-2cd. 


50  •       ELEMENTARY  ALGEBRA 

The  square  of  a  polynomial  is  equal  to  the  sum  of  the  squares 
of  each  terrro  inc7'eased  by  twice  the  product  of  each  term  with 
each  that  follows  it. 

The  student  should  note  that  the  square  of  each  term  is 
always  positive,  while  the  product  of  the  terms  may  have  the 
plus  sign  or  the  minus  sign. 

EXERCISE  26 

Find  by  inspection : 

1.  (^;c  +  y  +  zy.  7.  (2a'-3b'-S(y'y. 

2.  (m  +  n—py,  8.  {x  +  y-{-u-Sy. 

3.  (m-n-Sy.  9.  (2  a-2b-}-2  c-dy. 

4.  (m  +  2n-4)l  10.  (Sx-4ty -z  +  2uy. 

5.  (c?2_2e  +  l)2.  '     11.  (i^-2x'  +  3x-7y. 

6.  (2d-3e-4:fy. 

68.  In  simplifying  a  polynomial  the  student  should  remember 
that  a  parenthesis  is  understood  about  each  term.  Hence  after 
multiplying  the  factors  of  a  term,  the  beginner  should  inclose 
the  product  in  a  parenthesis. 

Ex.     Simplify  {x  +  6)(x  -  4)  -  {x  -3){x-  5). 

Check.     If  X  =  1, 
(X  +  6)(a:  -  4)  - (X  -  3)(x  -  5)  =  (7  .  -3) - (-2  .  -4)  =  -29. 
=  [a;2  +  2  X  -  24]  -  [ie2  -  8  X  +  15] 
=  x2  +  2  X  -  24  -  x^  +  8  X  -  15 
=  10  X  -  39.  =  10  -  39.  =  -  29. 

EXERCISE  27 
Simplify  the  following  expressions,  and  check  the  answers : 

1.  (x-7)(x-S)-(x-14:)(x-4:). 

2.  3{x-9)-5(x  +  7). 

3.  (a  +  7)(5a4-3)-2(a2-3a  +  7). 

4.  {a-{-by~{a-^by. 


MULTIPLICATION  51 

5.  (m  +  2)(m-2)  +  (m-7)(m-2)-3(m2-3m). 

6.  12(a  +  l)  +  17(a-2)-5(a-3)(a-2). 

7.  {d  +  e){d-e)-^{d-ef-{d  +  e)\ 

9.  (a  +  6)(a  +  c)-(6  +  c)(6  +  a)  +  (c  +  a)(c  +  6). 

10.  (a-26)2  +  (6-2c)2+(c-2a)2. 

11.  (x  +  2/)(^'-2/)  +  (2/  +  2;)(2/-2;)  +  (2;  +  x)(2J-a!). 

12.  (o  +  6  +  c)2-(a  +  6  +  c)(a-f  6-c). 

13.  a(a  +  6)-[(a-6)2  +  6(a-6)].  _    . 

14.  (a  -  6)2  +  (6  -  c)2  -f-  (c  -  a)2  _  (a  +  &  4-  c)^. 

15.  4.{x-4.y){x^4.y)-2{x-4.yy-2{x'  +  %y^). 

16.  4(a-6)(a  +  6)-4(a-2  6)(a-6)+12  5(a-6). 

17.  4 (a  +  6)  (?>  -  a)  -  3(6  -  a)  (a  4-  &)  -  a(a  -  6). 

18.  (5  a;  +  6  2/)^  —  (3  ic  —  7  2/)^  +  (4  a;  —  5  ?/)  (4  ic  +  5  2/). 

19.  (7aj-82/)'-(4«-2/)2-(3a;  +  42/)(3a;-42/). 

20.  (3a;-7)2-(5aj-2)2-(6a;  +  7)(7-6a;). 

21.  4(9p-8^)(9i>  +  8(?)-2(7jy-3g)2-(2p-5g)2. 

22.  3(a;-17)(a;-13)-4(a;-9)(a;-17)  +  (x-13)(a;-9). 

23.  (a; +  5)2 -(a; -9)2 -140. 

24.  5aj(3a?-ll)  +  2(4a;-7)2-75a;2-7(9  +  2a;)24-167. 

25.  (3 aa;  +  4 6)2+ (4 aa; -3  6)2- (5 aaj- 7  6)2  + 11 62. 

26.  9(5aa;-36)2-4(6aa;-76)2-(9aa;  +  56)(9aa;-56)- 
21  a6a;. 

27.  (2.5a;  +  3)(5-2a;)  +  (1.5-a;)(4  +  2a;)+7(a;-4)2. 

28.  {mn  +  1)^  —  3  mn  (mn  + 1)  • 

29.  (a;  +  l)(a;  +  2)(a;  +  3)-(a;  +  2)(a;  +  3)(aj  +  4)-3(a;  +  2) 
(aJ  +  3). 

30.  (p  +  2n)2-3(i)-5w)(i)  +  7n). 


52  ELEMENTARY  ALGEBRA 

31.  (a^  +  a^  +  x-j-l)(x-l)-(x'-}-l)(x  +  l)(x-l). 

32.  a  +  3[4-3(2-a)]. 

33.  5d-(6c?-5)  +  [6-4(d-9)]. 

34.  12  mn- 5  (2  m?i  -  7)  -  [3  mrt  -  4  (m?i  -  2)  -  (mn  -  2)]. 

35.  l(a  +  xy-(a-xy\-2\{a-{-x)(a-x)-{a-xy\. 

36.  (2x-l)(3x-\-5)(x  +  l)-(x-l)(x-2)(x-{-S). 

37.  (3a;  +  5)(2a;-3)(a^-l)-(a;-l)(ic4-2)(a^-3). 

38.  3  (a  +  2  6)  (a  -  6)  -  4  (2  a  -  3  6)  (3  a  +  &)+  &  (2  a  -  9  6) . 

39.  x-(9a-\-Sb-7c)  (f+g)  +  (10  a  +  8  6  -  7  c)  (/-^). 

40.  (3a-6c)(4:a-3d)-\{2a-5c)((ya-lld)-{S7cd- 
6ac)\. 

41.  From  0  subtract  three  times  the  difference  between  the 
squares  of  (a  +  b)  and  (a  —  b)  and  add  the  result  to  6  a6. 


CHAPTER   IV 
DIVISION 

69.  Division  is  the  process  of  finding  one  of  two  factors  if 
their  product  and  the  other  factor  are  given. 

The  dividend  is  the  product  of  the  two  factors,  the  divisor 
is  the  given  factor,  and  the  quotient  is  the  required  factor. 

Thus  to  divide  —  12  by  +  3,  we  must  find  the  number  which  multi- 
plied by  +  3  gives  —  12.     But  this  number  is  —  4  ;  hence  ^^^^^-  =  —  4. 

+  3 


70.   Since 

+  a-+6  =  +  a& 

-f  a  •  —  6  =  —  a6 

—  a-+b  =  —  ab 

and 

^a'  —  b  =  -{-ab, 

it  follows  that . 

+  a 

-«*=  b 
+  a 

:z«^  =  +  b 

—  a 

+"*=  b. 

—  a 

71.  Hence  the  law  of  signs  is  the  same  in  division  as  in 
multiplication  :  Like  signs  produce  plus,  unlike  signs  minus. 

72.  Law  of  Exponents.     It  follows  from  the  definition  that 
a^-^a^=z  a%  for  a^  xa^  =  a\ 

Or  in  general,  if  m  and  n  are  positive  integers,  and  m  is 
greater  than  n,  a"' -i- a"  =  a"' -  ",  for  a"'-''a"  =  a"*. 

53 


54  ELEMENTARY  ALGEBRA 

The  exponent  of  a  quotient  of  ttvo  powers  with  equal  bases 
equals  the  exponent  of  the  dividend  diminished  by  the  exponent 
of  the  divisor. 

DIVISION  OF  MONOMIALS 

73.  To  divide  10  x^y^z  by  —  2  x^y^,  we  have  to  find  the 
number  which  multiplied  by  —  2  a^/  gives  10  x^yh.  This 
number  is  evidently  —  5  oi^yz. 

Therefore,  ^^f^^^^-Bx^yz. 

—  2xfy^ 

Hence,  the  quotient  of  two  monomials  is  a  monomial  whose 
coefficient  is  the  quotient  of  their  coefficiemts,  preceded  by  the  proper 
sign,  and  whose  literal  part  is  the  quotient  of  their  literal  parts 
found  in  accordance  with  the  law  of  exponents. 

EXERCISE  28 

Perform  the  divisions  indicated: 

1.    _64---4.  ^    98  pV^  ,^    96mVia^ 

o.     •  15. 


2.    64 ---4. 


4. 


1  q  12  m'^n^^x 

g     -20aVV  -144aW 

44  a  ^  ■     -20aV  ib.   _— -j^-^. 

-11«*  10.   l^i^^^l^.  ^^      50  dV/ 

-17a62  -limp  i^-    _5o^4/ 


(^-&) 


lib'  ^^    («±&Z. 

{a  +  bf  18. 

-7a^z/2«  19.      6(a4-6-cy  , 

e.   ZlI^^.  ^^    28  mV/  -2(a-|-5-c)« 

15ac«  13.    ^^e^T^r  ^^     40  r/y  . 

^     -12a^/y^  ^^     -16mpY.  ^ 

—  3  x^yz"^  '    — 16  mpV 


DIVISION  55 


DIVISION  OF  POLYNOMIALS   BY  MONOMIALS 

74.  To  divide  ax-\-bx-\-  ex  by  x  we  must  find  an  expression 
which  multiplied  by  x  gives  the  product  ax-\-bx-{-  ex. 

■^^*  £c(a  +  6  4-  c)  =  ax  -{-bx  +  ex. 

XT                        ax-\-bx-\-cx  ,  ,   , 

Hence  ' =  a-\-b  -{-c. 

X 

To  divide  a  polynomial  by  a  monomial,  divide  each  term  of  the 
dividend  by  the  monomial  and  add  the  partial  quotients  thus 
formed. 

E.g.      ^ — L_X ^—=z2y?yz^  -\-hyz-l. 

—  3  xyz^ 

EXERCISE  29 

Perform  the  operations  indicated : 

^^ ax  —  63  bx  4a;^  —  3a^  +  5a; 

Ix  '  —X 

2  -26aV  +  39a^(Z^  -9a^  +  6a^-3a9 

13a«  *  *  -3a^ 

3  -14a^6V4-21a^6V  - 11  a^y  4- 22  arV  -  33  a^y^ 

-la'V'e^  '  '  11  xy 

4aV-40aV4-8aV 
4a=^ 

g     -  12  a^6V  +  6  a^6V  -  18  a^b^c" 

ea'b'e^ 
g     25p^-20p*-5p\ 


-5p^ 
10. 


49  m?r?x  —  28  mVy  +  7  m'*n^ 


-7mV 

^j     169  fl^/g^  -  26  g^y^^  +  39  ^fz^ 
13a^?/V 

12.  (92  r^^  - 115  r^s*  - 161  i^s^  +  69  rV)  -5-  23  rs. 

13.  (187  a^y  - 121  aj3^  -  88  a^/  +  231  a^^T^  -•  11  xy. 


56  ELEMENTARY  ALGEBRA 

14e    (75  ab'  -  105  cC~h  -  165  a%^  +  180  ah)  ^  15  ah. 
15.    (6.8  ah  —  8.5  ac  —  5.1  ad  -\-  3.4  ae)  -^  1.7  a. 

^g     6(a  +  6)«  +  15(a  +  &)'  ^g  a«»4-a^"  +  ft" 
-3(a  +  &)'          °               *  a" 

^^     4  (a; -:?/)- 8  (a; -y)^  ^^  a^y  +  x^"'?/"  +  a;^* 
—  2  (a?  —  2/)                            *  —  a;"*2/" 

/oU.     : • 


DIVISION  OF  A  POLYNOMIAL   BY  A  POLYNOMIAL 

75.  Let  it  be  required  to  divide  25  a  —  12  +  6  a^  —  20  a^  by 
2  a^  +  3  —  4  a,  or,  arranging  according  to  descending  powers  of 
a,  divide     6a3-20a2  +  25a-12  by  2a2_4a  +  3. 

The  term  containing  the  highest  power  of  a  in  the  dividend  {i.e.,  6a^)  is 
evidently  the  product  of  the  terms  containing  respectively  the  highest 
power  of  a  in  the  divisor  and  in  the  quotient. 

Hence  the  term  containing  the  highest  power  of  a  in  the  quotient  is 
i«!,  orSa. 

If  the  product  of  3  a  and  2  a2  _  4  q^  4.  3^  ^-.g.  6  «3  _  12  a^  +  9  a,  be  sub- 
tracted from  the  divisor,  the  remainder  is  —^d^-\-\%a  —  12. 

This  remainder  obviously  must  be  the  product  of  the  divisor  and  the 
rest  of  the  quotient.  To  obtain  the  other  terms  of  the  quotient  we  have 
therefore  to  divide  the  remainder,  —  8  a^  4-  I6  «  —  12,  by  2  a^  _  4  ^5  +  3. 

We  consequently  repeat  the  process.  By  dividing  the  highest  term  in 
the  new  dividend  —  8  a^  by  the  highest  term  in  the  divisor  2  c^^  we  obtain 

—  4,  the  next  highest  term  in  the  quotient. 

Multiplying  —  4  by  the  divisor  2  a^  _  4  qj  4.  3  we  obtain  the  product 

—  8  a^  4. 16  05  _  12,  which  subtracted  from  the  preceding  dividend  leaves 
no  remainder. 

Hence  3  a  —  4  is  the  required  quotient. 
The  work  is  usually  arranged  as  follows  : 


6  a3  _  20  a2  +  25  «  -  12 

2  a2  _  4  05  +  3 

6a3_i2a2^9a 

3«-4 

-  8a2+i6a_l2 

-  8  a2  +  16  a  -  12 

DIVISION  57 

76.  The  method  which  was  applied  in  the  preceding  example 
may  be  stated  as  follows : 

1.  Arrange  dividend  and  divisor  according  to  ascending  or 
descending  powers  of  a  common  letter. 

2.  Divide  the  first  term  of  the  dividend  by  the  first  term  of  the 
divisor,  and  write  the  result  for  the  first  term  of  the  quotient. 

3.  Multiply  this  term  of  the  quotient  by  the  whole  divisor ^  and 
subtract  the  result  from  the  dividend. 

4.  Arrange  the  remainder  in  the  same  order  as  the  given 
expression,  consider  it  as  a  new  dividend  and  proceed  as 
before. 

5.  Continue  the  process  until  a  remainder  zero  is  obtained,  or 
until  the  highest  power  of  the  letter  according  to  which  the  dividend 
ivas  arranged  is  less  than  the  highest  power  of  the  same  letter  in 
the  divisor. 

77.  Checks.  Numerical  substitution  constitutes  a  very  con- 
venient, but  not  absolutely  reliable  check. 

An  absolute  check  consists  in  multiplying  quotient  and 
divisor.  The  result  must  equal  the  dividend  if  the  division 
was  exact,  or  the  dividend  diminished  by  the  remainder  if  the 
division  was  not  exact. 

Ex.  1.     Divide  8a'^  +  8a-4  +  6a*-lla2  by  3a-2. 
Arranging  according  to  descending  powers, 


6a*  +  8a3-lla2  +  8a-4 
6  a*  -  4  a3 


4-12a3-lla2 
+  12  a8  -  8  a2 

-  3a2  +  8a 

-  3a2  +  2a 

-f6a- 

-4 
-4 

Check.    Ifa  =  6  =  l, 
Sa-2  =7-1 


2a3-f  4a2_a4.2  =7= 


58  ELEMENTARY  ALGEBRA 

Ex.  2.     Divide  a^  -  4  6^  -  6  a«6  +  9  a'b'  hj2b''-Sah+  a\ 

Arranging  according  to  descending  powers  of  a,  we  have 

a*  -  6  «36  +  9  a252  _  4  54  1  a2-Saj)  +  2b^ 

g*  -  3  a^  +  2  g^^^  |  g^  -  3  g6  -  2  &2 

-  3  g3&  4-  9  g262  _  6  ab^ 


-  2  g2&2  +  6  g63  _  4  54 
-2a2&2  +  6g63_4  6* 


Check.  The  numerical  substitution  g  =  1,  6  =  1,  cannot  be  used  in  this 
example  since  it  renders  the  divisor  zero.  Hence  we  have  either  to  use 
a  larger  number  for  g,  or  multiply. 

(g2  -  3  a6  +  2  62)(a2  _  3  «5  _  2  62) 
=  [(g2  -  3  g6)  +  2  62] [(g2  _  3  g6)  -  2  62] 
=  (g2_3g6)2-4  6* 
=  g2  -  6  g36  +  9  g262  -  4  6*. 

Ex.3.    Divide  a^H- 6^  + c^ -3a6c  by  a  +  &  +  c. 

Check.     If  g  =  6  =  1, 

g3-3g6c  +  68H-c8  |g  +  6  +  c =0-3 

gs  +  g26     +  g2c  \a^-ab-ac+  b-  -  6c  +  c2       =0. 


-a^b- 
-a^b- 

-ab^ 

-  3  g6c  +  63  +  ( 
-g6c 

.3 

- 

-  a^c 
-g2c 

+  g62  - 

2  g6c  +  68  4-  c8 
g6c  -  gc2 

g62- 
g62 

g6c  4- 

gc2  4-63   +  c8 
4-63   +62c 

g6c  4- gc2  -  62c  +  c3 
.    a6c           _62c-6c2 

gc2  4-  6c2  4-  c3 
gc2  4-  6c2  +  c8 

Ex.  4.   Divide  a"'+3  —  2  a'"+^  —  a"*-^  by  a'^+i  -\-2a'^-^  a'^-\ 

g»»+3  —  2  g'»+i  4-  g"*-!  I  qm+i  .4.  2  g"*  4-  g"*-! 

qm+3  _!■  2  a*»+2  ^  qm+1 |  g2  -  2  g  4-  1 

—  2  g"»+2  _  3  ^m+l  4.  ^m-l 

—  2  g'»+2  -  4  g'^+i  -  2  g« 


am+\  _j_  2  g"*  4-  g"*"^ 
a">+i  _|_  2  gw*  +  g"*-! 


DIVISION  69 

EXERCISE  30 
Divide : 

1.  ic2_7a;  +  12  by  x-A.. 

2.  a^-a-n  by  a-9. 

3.  6a2  +  23a6  +  2062  by  2a  +  5&. 

4.  1562_196c-56c2  by  36-8c. 

5.  42a^-10aJ2/-12/  by  7ic  +  32/. 

6.  4a2-12162  by  2a-ll&. 

7.  21a2  4-13a&-20&2  by  7a-55. 

8.  16r2-46rs  +  15s2  by  2r-5s. 

9.  40%2_53^^_l_6y2  |3y  5^_g^ 

10.  18p2  +  I9^g  - 12  q"  by  2i)  +  3  g. 

11.  90  m^  -  281  mn  -  85  n^  by  5  m  -  17  n. 

12.  72^2  _^  7^^  _  2  g2  by  ^p  -  q. 

13.  aj^-l-?/^  \,j  x  +  y. 

14.  a;^  —  2/^  by  a;  —  y. 

15.  2a3_9a2  +  lla-3  by  2a-3. 

16.  13p  4- 1  +  47y  +  35y  by  5^)  + 1. 

17.  4a3-24a-9-3a2  by  a-3. 

18.  2a3  +  18-3a-7a2  by  2a  +  3. 

19.  a^  —  16y^hjx  —  2y. 

20.  9o2  +  24a6  +  1662_36c2  by3a  +  4&  +  6c. 

21.  16a^-40a;y  +  25/-922  by  4a;-52/-32. 

22.  16  +  40a  +  25a2-49a^  by  4  +  5a-7a2. 

Perform  the  operations  indicated  and  check  the  answers : 

23.  (216  a^  4- 125)  -  (36  a^  _  30  ^^  4-  25). 

24.  (128  a'b^  - 160  a'b'  +  2a%  +  15  a')  -^-  (3  a^  -  8  ab). 

25.  (1  -  32 p') ^ (1  +  2i)  +  4p2  +  Sp'  + 162>0. 


60  ELEMENTARY  ALGEBRA 

26 .  (81  xy  +  72 /  -  63  xy""  +  56y -i9xy -\-63  x'y) 

-i-(^x^  -\-S  —  7  x). 

27.  (1  4- 81  a^-18a0^(l- 6a +  9a2). 

28.  (15 X*  +  7 X  +  7 x^  +  15 x^  +  4:)-ir{S X'  +  2 X  +  1). 

29.  (7  a^  ^  2  0^4  +  82  aj2  +  145  X  +  72)  -  (9  +  8  a;  -  a^). 

30.  (15 a^-34.a^-^a^-\-2a-S)-^{3(v'-5a- 4). 

31 .  (42  m^  +  174  m^  +  70  -  338  m  +  4  m^)  --  (3  m^  +  3  m  -  7) 

32.  (lSp'-9pq'-33phj-i-S2pY-Sq')-^(6p^-7pq-\-Sq'). 

33.  (56a'  +  119  a^ft^ 4. 45  6^-53  a^b-47  aWy---(7  d'+^  6^-4  ab). 

34.  (x^-f)^(x-y).  36.    (a*_?)4)-f-(a4-6), 

35.  (8a;^-27/)-^(2a^-3?y).      37.    (a' +  b')-i-(a  +  b). 

38.  (81a46*-625c^)--(3a6-5c). 

39.  (a^  +  a262  +  64)--(a2-a6  +  62). 

40.  (4a*-13a262  +  9  6*)-^(2a2  +  afe-3  62). 

41.  (a^  +  3a26  +  3a&2_^63  +  c«)H-(a  +  6  +  c). 

42.  (a'-3ab  +  b^  +  l)-^(a  +  b-\-l). 

43.  (a2_6a&  +  9  62-c2-2c(^-d2)--(a-3  6-c-d). 

44.  (a3-3a26  +  3a62-?/-l)^(a~6-l). 

45.  (a^  +  b^-3ab  +  l)-^(a'  +  b^  +  l-ab-a- b). 

^^    a^r._y3rn  ^^    6  ci"^  +  23  a'^-'b  +  20  a'^-^b^ 

■    ^m_ym'  •  2  a"'-^ -{- 5  a'^-^b 

x'"^  +  a;'^"'?/^"  +  y^_  gQ    1.4  X-—  63.08  a;y  +  3.6  y\ 

x^""  +  x'^y''  +  2/^"  *  "  7  x  —  Ay 

48    a"^+^  +  3  a"^+^  +  3  a"+^  +  ft",    gj     .3p^  + .03pg- .06g^ 
a«'+i  4- 2  a"*  4- a'""^        *        "  .5p-.2q 

52.    (i>'--ii2M-i?')-(fP  +  g). 
Find  four  terms  of  the  quotient : 

53.    -i-.  54.    -J— . 

14-a;  1— a; 


DIVISION  61 


SPECIAL   CASES  IN  DIVISION 

78.   Division  of  the  difference  of  two  squares. 

Since  (a  +  h)  (a  -h)  =  a^  -  h\ 

=  a  +  0;  and  =ia  —  o. 

a—b  a-]-b 

I.e.  the  difference  of  the  squares  of  two  numbers  is  divisible 
by  the  difference  or  by  the  sum  of  the  two  numbers. 

Ex.1,    9^!^iM  =  3x  +  4  8,. 
3x  -  4y 

Ex.  2.    (0^  +  ^)'-  {^-yy=ra  ^b)  -  (x-y)  =  a +  b -x  +  y. 
(a  +  6)  +  (x  -  y) 

EXERCISE  81 

Write  by  inspection  the  quotient  of : 


1 

a'-l 

a-1 

2. 

b'-4: 

b-\-2 

3. 

C'-4:CP 

c-2d 

A 

^x'-9f 

K 

49ajy-64^2 

Ix'y'-Sz 

6. 

121  x'yh'-ia'b' 

11  xyz  +  2  ah 

7. 

25a«-l 

5a^  +  l 

Q 

(a4.6)2_4aj* 

2x-\-^y  (a  +  6)-2ar 

Find  exact  binomial  divisors  of  each  of  the  following  ex- 
pressions : 

9.    a^-b\  15.  ima^-h''^. 

10.  a;^-/.  16.  a?'^-b^. 

11.  a^-2/^  17.  (a-\-bf-22op\ 

12.  ai2_6i2,  18.  (a -6)^" -25^1"". 

13.  <y«-625.  19.  (a  +  6  +  c)2-l. 

14.  289a«6«-196. 


62  ELEMENTARY  ALGEBRA 

Find  two  factors  of : 

20.  10,000-81.  22„   999,991. 

21.  8099.  23.   63.91. 

79.    Division  of  the  sum  or  the  difference  of  two  cubes. 

By  actual  division,  we  obtain 

'''-^'  =  a'  +  ab  +  b' 


a  —  b 


and 


a^  +  b^       2        J.  ,  -wi 


a^b 

The  diiference  of  the  cubes  of  two  numbers  divided  by  the 
difference  of  the  two  numbers  equals  the  square  of  the  first,  plus 
the  product  of  the  first  and  the  second,  plus  the  square  of  the  second. 

The  sum  of  the  cubes  of  two  numbers  divided  by  the  sum  of  the 
two  numbers  equals  the  square  of  the  first,  minus  the  product  of 
the  first  and  the  second,  plus  the  square  of  the  second. 

EXERCISE  32 
Write  by  inspection  the  quotient  of : 

1.  (a?-y^)^{x-y).  5.    (m^- 125)-j-(m- 5). 

2.  (x^J^f)-^{x  +  y).  6.    (21a^  +  W)^(;da  +  h). 

3.  (c3  +  l)--(c4-l).  7.    (64a«+276«)--(4a+36). 

4.  (mV-l)--(m7i-l).  8.    (a^ -b^)~{a^ -b^). 

EXERCISE   33 

Determine  what  binomial  or  binomials,  if  any,  will  divide 
each  of  the  following  expressions,  and  find  the  quotient  or 
quotients : 

1.  m-  +  8.  4.   0^2/3  _|_  27.  7.   343 -a.y. 

2.  m3-8.  5.    a^-64.  8.    512  +  aj». 

3.  a?f-21.  6.    a^W  +  c\  9.    3?  +  y\ 


DIVISION  63 

10.  729a^-y^.  14.    a;«  +  121.  18.  lOOOa^y  +  ^s. 

11.  x^-\-125.  15.    x''-y\  19.  216  a^- 8  61 

12.  0^6-125.  16.    a^-f.  20.  (x  +  2/)'-8;23^ 

13.  a;«-121.  17.    a^-lfc\  21.  ««  +  &'. 

Find  numerical  divisors  of  the  following : 

22.  1000-64.  24.    1001.  26.  8027. 

23.  1000  +  27.  25.    1,000,001.  27.  64,000,001. 


CHAPTER  V 
LINEAR   EQUATIONS   AND   PROBLEMS 

80.  The  first  member  or  left  side  of  an  equation  is  that  part  of 
the  equation  which  precedes  the  sign  of  equality.  The  second 
member  or  right  side  is  that  part  which  follows  the  sign  of 
equality. 

Thus,  in  the  equation  2a;  +  4  =  aj  —  9,  the  first  member  is  2 x  +  4,  the 
second  member  is  a;  —  9. 

81.  An  identical  equation  or  identity  is  an  equation  which  is 
true  for  all  values  of  the  letters  involved. 

Thus,  {a  +  &)(«  —  &)=  a^  —  6^,  no  matter  what  values  we  assign  to  a 
and  h.  The  sign  of  identity  sometimes  used  is  =,  thus  we  may  write 
(a  +  6)  (a  ~  6)=  a^  -  62. 

82.  An  equation  of  condition  is  an  equation  which  is  true  only 
for  certain  values  of  the  letters  involved.  An  equation  of  con- 
dition is  usually  called  an  equation. 

a;  4-  9  =  20  is  true  only  when  ic  =  11 ;  hence  it  is  an  equation  of 
condition. 

83.  A  set  of  numbers  which  when  substituted  for  the  letters 
in  an  equation  produce  equal  values  of  the  two  members,  is 
said  to  satisfy  an  equation. 

Thus  x=\2  satisfies  the  equation  x  +  1  =  13.  x  =  20,  y  =  7  satisfy 
the  equation  cc  —  ?/  =  13. 


84.  An  equation  is  employed  to  discover  an  unknown  number, 
usually  denoted  by  x,  ?/,  or  z,  from  its  relation  to  known 
numbers. 

64 


LINEAR  EQUATIONS  AND  PROBLEMS  65 

85.  If  an  equation  contains  only  one  unknown  quantity,  any 
value  of  the  unknown  quantity  which  satisfies  the  equation  is 
a  root  of  the  equation. 

^  is  a  root  of  the  equation  2x  +  2  =  20. 

86.  To  solve  an  equation  is  to  find  its  roots. 

87.  A  numerical  equation  is  one  in  which  all  the  known  quan- 
tities are  expressed  in  arithmetical  numbers ;  as  (7  —  x)(x-\-  4) 
=  a^-2. 

88.  A  literal  equation  is  one  in  which  at  least  one  of  the 
known  quantities  is  expressed  by  a  letter  or  a  combination  of 
letters;  as  x  -\-  a  =  bx  —  c. 

89.  A  linear  equation  or  an  equation  of  the  first  degree  is  one 
which  when  reduced  to  its  simplest  form  contains  only  the 
first  power  of  the  unknown  quantity ;  as9a;  —  2  =  6ic  +  7. 
A  linear  equation   is  also  called  a  simple  equation. 

90.  The  process  of  solving  equations  depends  upon  the  fol- 
lowing principles,  called  axioms : 

1.  If  equals  be  added  to  equals,  the  sums  are  equal. 

2.  If  equals  be  subtracted  from  equals,  the  remainders  are  equal. 

3.  If  equals  be  multiplied  by  equals,  the  products  are  equal. 

4.  If  equals  be  divided  by  equals,  the  quotients  are  equal. 

5.  Like  powers  or  like  roots  of  equals  are  equal. 

Note.  Axiom  4  is  not  true  if  the  divisor  equals  zero.  E.g.  0x4 
=  0x6,  but  4  does  not  equal  5. 

91.  Transposition  of  terms.  A  term  may  be  transposed  from 
one  member  to  another  by  changing  its  sign. 

Consider  the  equation  x  -\-  a  =  b. 

Subtracting  a  from  both  members,  x  =  b  —  a.         (Axiom  2) 
I.e.  the  term  a  has  been  transposed  from  the  left  to  the  right 
member  by  changing  its  sign. 
Similarly,  ii  x  —  a  =  b. 
Adding  a  to  both  members,  x=b  -\-  a.  (Axiom  1) 


66 


ELEMENTARY  ALGEBRA 


The  result  is  the  same  as  if  we  had  transposed  a  from  the 
first  member  to  the  right  member  and  changed  its  sign. 
It  follows  from  §  91  that : 

92.  Any  term  that  occurs  ivith  the  same  sign  in  both  members 
of  ayi  equation  may  he  canceled. 

93.  Tlie  sign  of  every  term  of  an  equation  may  be  changed 
without  destroyiiig  the  equality. 

Consider  the  equation  — x-\-a=—h-^c. 

Multiplying  each  member  by  —  1,     x  —  a  =  h  —  c.  (Axiom  3) 


SOLUTION   OF  LINEAR  EQUATIONS 

94.   Ex.  1.     Solve  the  equation  6a;  —  5  =  4a;  +  l. 

Transposing  4  x  to  the  first,  and  5  to  the  second  member, 

6x-4ic  =  H-  5. 
Uniting  similar  terms,  2  x  =  6. 

Dividing  both  members  by  2,  x  =  3.  (Axiom  4) 

Check.     Whenic  =  3. 

The  first  member,  6  x  -  5  =  18  -  5  =  13. 

The  second  member,  4  x  +  1  =  12  +  1  =  13. 

Hence  the  answer,  x  =  3  is  correct. 

Ex.  2.     Solve  the  equation  4  -  3  (a;  -  2)  =  6  (a^+  3)  -  6  a;. 

Simplifying  both  members,       4  —  3x  +  6  =  6x  +  18  —  6x. 
Transposing, 
Uniting  terms, 
Dividing  by  —  3, 

Cheek.    If  x  =  —  f . 

The  first  member,  4  -  3(x  -  2)  =  4  -  3(-  J^)  =  4  +  14  =  18. 

The  second  member,   6  (x  +  3)  -  6  x  =  6  (|)  -  6(-  f)  =  2  +  16  =  18. 


-  3x-6x  +  6x  =  -4-6  +  18. 
-3x  =  8. 


LINEAR  EQUATIONS  AND  PROBLEMS  67 

95.  To  solve  a  simple  equation,  transpose  the  xmknown  terms 
to  the  first  member,  and  the  known  terms  to  the  second.  Unite 
similar  terms,  and  divide  both  members  by  the  coefficient  of  the 
unknown  quantity. 

Ex.  3.     Solve  the  equation  (4  -  ar)  (5  -  a;)  =  2  (11  -  3  «)  +  ar^. 

Simplifying,  20  -  9  a;  +  a;^  =  22  -  6  a;  +  x^. 

Canceling  ic^  and  transposing,  —  9  x  +  6  a;  =  —  20  +  22. 

Uniting,  -  3  a;  =  2. 

Dividing  by  —  3,  a;  =  —  |. 

Check.    Ifx=-|. 

The  first  member,  (4  -  a;)  (5  -  x)  =  (4  + 1)  (5  +  |)  =  ^  .  V  =  -F  =  26^ 

The  second  member,  2  (11  -  3  x)  +  x2  =  2  (11  +  2)  +f  =  26  +  |  =  26|. 

Ex.4.     Solve  the  equation  1.3  aj- [.3 -.5  (a; -3)]  =3.8 -a;. 

Simplifying,  1.3  x  -  [.3  -  .5  x  +  1.5]  =  3.8  -  x, 

or,  1.3x- .3  + .5x- 1.5  =  3.8-x. 

Transposing,  1.3  x  +  .5  x  +  x  =  .3  +  1.5  +  3.8. 

Uniting,  2.8  x  =  5.6. 

Check.    If  X  =  2.  a:  =  2. 

The  first  member,    1.3  x  -  [.3  -  .5  (x  -  3)]  =  2.6  -  [.3  -  .5  (-  1)] 

=  2.6 -.8  =  1.8. 
The  second  member,  3.8  -  x  =  3.8  -  2  =  1.8. 

Note.  The  decimals  in  Ex.  4  could  be  removed  by  multiplying  each 
member  by  10. 

Ex.  5.     Solve  the  equation  |(a;  —  4)  =  J  (a;  -j-  3). 

Simplifying,  \x  —  2  =  \x-\-\. 

Transposing,  ^x  —  |^x  =  2  +  l. 

Uniting,  |  x  =  3. 

Dividing  by  |,  x  =  18. 

Check.     If  X  =  18. 

The  left  member  K«  -  4)  =  i  x  14  =  7. 

The  right  member  ^x  +  3)  =  |  x  21  =  7. 

Note.  Instead  of  dividing  by  \  both  members  of  the  equation  |  x  =  3, 
it  would  be  simpler  to  multiply  both  members  by  6. 


9. 

69-7a;  =  99-13a;. 

10. 

5x  +  S6-9x  =  12. 

11. 

9aj-23  =  13-3a;. 

68  ELEMENT AliY  ALGEBRA 

EXERCISE  34 

Solve  the  following  equations  and  check  the  answers : 

1.  Sx  =  ll-\-2x.  8.   14x-12a;  =  10-3a;. 

2.  5x  =  3x  +  12. 

3.  2a;  =  10-3x. 

4.  7  a; +  16  =  30. 

5.  25  +  3.T  =  46.  12.    lla;-72  +  9x  =  88  +  10a; 

6.  4a;-29  =  ll.  13.   4 aj-27  =  1 -3 a;. 

7.  16a;-7  =  15x-3.  14.   24-3a;  =  25-4a;. 

15.  25  +  6i»-8a7  =  17-4a;  +  12. 

16.  20-7a;H-9  =  40-6a?  +  30. 

17.  1  x  =  m-(12x-\-21). 

18.  6  ic  -  (11  X- 10)  =  87 -(21 +  13  a;). 

19.  10a;-(4a;  +  2)  =  6-(21a;-19). 

20.  7(8  a? -5)  =  77. 

21.  5(2  aj  +  7)  =  9(2a;- 5). 

22.  6(3a;-8)  =  7(5aj-19). 
.  23.  5aj  +  7(2a;  +  3)  =  59. 

24.  13aj-2(4a;-5)  =  66-3aj. 

25.  6(5aj-2)-5(6a;-5)  =  4(9-2a;)+l. 

26.  7(6  a;  +  5)  -  8(3  -  4  a;)  +  24  =  12(2  a;  +  3)  + 199. 

27.  a;2  +  13a;-(aj2-a;)  =  5(2aj  +  3)  +  5. 

28.  a;(a;  + 10) -6(aj- 5)  =  4- (a; -ar^). 

29.  (a;-l)(a;  +  6)  =  (a;  +  5)(a;-2). 

30.  (a;  -  5)(a;  -  7)  =  (a; +  4)(a;- 9) -13. 

31.  (a;+3)(a;-6)  =  (a;-5)(a;  +  6)-20. 

32.  (a;  +  9)(a;  +  2)  =  (a;-3)(a;  +  l)  +  60.' 

33.  (a;-13)(a;-5)  =  (a;-10)(aj-ll). 


LINEAR  EQUATIONS  AND  PROBLEMS  69 

34.  (2x  +  2)(4.x-S)  =  (4.x-4)(2x-\-S). 

35.  {3x-5)(2x  +  5)-(x-{-l)(6x-4t)  =  0. 

36.  (4  a;+l)(7  a;  +  4)  -  (14  a;  + 1)(2  a;  - 1)  =  285. 

37.  (a;  +  5)2 -(aj- 9)2  =  140. 

38.  (x  -  If  +  {x  +  4)2  =  {x-  3)2  +  (x  +  5)2  +  11. 

39.  {x  +  4)2  -{x-  6)2  =  (a;  +  2)2  -  (x  +  3)2  + 161. 

40.  2(a;  +  l)2-<2a!-3)(aj  +  2)  =  12. 

41.  4(a;-l)(a;-2)-2(aj-7)(2a^  +  l)-6  =  0. 

42.  (5aj-2)(a;-3)-5(a;-l)(a;-4)  +  14  =  0. 

43.  (4  2/-7)(9  2/-48)-12(3  2/  +  l)(2/-6)=0. 

44.  a;(a;-l)(a;  +  7)  =  (a;  +  l)(a;  +  2)(a;  +  3). 

45.  22a;-[3-(3a;-2)]  =  2a;-(4a;-3). 


46. 

\x-2  = 

ia^  +  3. 

47. 

Jx  +  iaj 

=  15. 

48. 

\x  +  l^-. 

=  ia^-17. 

49.  ^x-\-\x-\-\x-\-\x==l, 

50.  \{x-4.)  =  \(x^2), 

51.  19-(17+|-a;)=ia;  +  7. 

52.  4(|a5-3)-y%a;  =  |(2aj-16)-6. 

53.  8.7  a? +  6.8  =  5  a; +  7.6 +  3.8  a;. 

54.  .38  a; -.18  a; -(.93 -.29  a;)  =  .21 -(.41  a; -.28). 

55.  a^-  (.2a;  +  3)(3a;+  .4)  =  (4 a;  +  .7) (.la;.-  3)  -1.95. 

SYMBOLICAL  EXPRESSIONS 

96.  Suppose  one  part  of  70  to  be  x,  and  let  it  be  required  to 
find  the  other  part.  If  the  student  finds  \t  difficult  to  answer 
this  question,  he  should  first  attack  a  similar  problem  stated 


70  ELEMENTARY  ALGEBRA 

in  aritlimetical  numbers  only,  e.g. :  One  part  of  70  is  25 ;  find 
the  other  part.  Evidently  45,  or  70  —  25,  is  the  other  part. 
Hence  if  one  part  is  x,  the  other  part  is  70  —  x. 

Whenever  the  student  is  unable  to  express  a  statemetit  in  alge- 
braic symbols,  he  should  formulate  a  similar  question  stated  in 
anthmetical  numbers  07ily,  and  apply  the  method  thus  found  to  the 
algebraic  problem, 

Ex.  1.  What  must  be  added  to  a  to  produce  a  sum  6? 

Consider  the  arithmetical  question :  What  must  be  added  to  7  to  pro- 
duce the  sum  12  ? 

The  answer  is  6,  or  12  —  7. 

Hence  h  —  a  must  be  added  to  a  to  give  h. 

Ex.  2.   x-\-y  yards  cost  $100;  find  the  cost  of  one  yard. 

If  7  yards  cost  one  hundred  dollars,  one  yard  will  cost 

Hence  \ix  +  y  yards  cost  $  100,  one  yard  will  cost  dollars. 

x-\-y 

EXERCISE  35 

1.  By  how  much  does  7  exceed  a  ? 

2.  By  how  much  is  5  6  greater  than  3  5? 

3.  What  number  exceeds  a  by  7  ? 

4.  What  is  the  4th  part  of  a;  ? 

5.  What  is  the  wth  part  of  a.  ? 

6.  By  how  much  does  the  third  part  of  a  exceed  the  fourth 
part  of  6  ? 

7.  By  how  much  does  the  double  of  a  exceed  the  tenth  part 
of  6? 

8.  Two  numbers  differ  by  9,  and  the  smaller  one  is  n.  Find 
the  greater  one. 

9.  Divide  20  into  two  parts  such  that  one  part  equals  x. 
10.   Divide  a  into  two  parts  such  that  one  part  is  7. 


LINEAR  EQUATIONS  AND  PROBLEMS  71 

11.  What  is  the  dividend  if  the  divisor  is  a  and  the  quotient 
is  6? 

12.  What  is  the  quotient  if  the  dividend  is  x-\-y  and  the 
divisor  is  m  ? 

13.  What  number  divided  by  10  will  give  a;  as  a  quotient? 

14.  By  what  number  must  x-{-yhe  divided  to  give  10  as  a 
quotient  ? 

15.  What  is  the  dividend  if  the  divisor  is  d,  the  quotient  is 
g,  and  the  remainder  is  r  ? 

16.  Divide  a  into  two  parts  such  that  one  part  is  b. 

17.  How  much  does  a  lack  of  being  b  ? 

18.  What  is  the  excess  of  a  + 12  over  6  + 12  ? 

19.  What  is  the  excess  of  a  +  6  over  b-\-c? 

20.  The  difference  between  two  numbers  is  d  and  the  smaller 
one  is  x.     Find  the  greater  one. 

21.  What  number  must  be  subtracted  from  2  a;  +  4,  to  give  a 
remainder  3x  —  5? 

22.  A  is  a;  years  old  and  B  is  i/  years  old.     How  many  years 
is  A  older  than  B  ? 

23.  A  is  a;  years  old.     How  old  was  he  7  years  ago  ?     How 
old  will  he  be  y  years  hence  ? 

24.  If  A's  age  is  x  years,  and  B's  age  is  y  years,  find  the  sum 
of  their  ages  6  years  hence. 

25.  X  exceeds  an  unknown  number  by  y.    Find  that  number. 

26.  Two  numbers  differ  hy  x-\-y,  and  the  greater  one  is  3  y. 
Find  the  smaller  one. 

27.  A  product  consisting  of  two  factors  equals  a.     Find  one 
of  the  factors,  if  the  other  equals  2  x. 

28.  The  smallest  of  3  consecutive  numbers  is  x.    What  are 
the  other  two  ? 


72  ELEMENTARY  ALGEBRA 

29.  The  greatest  of  four  consecutive  numbers  is  y.  Find 
the  other  numbers. 

30.  A  has  m  dollars,  and  B  has  y  dollars.  If  A  gives  B 
4  dollars,  find  the  amount  each  will  then  have. 

31.  How  many  cents  are  in  a  dollars  ?     In  6  dimes  ? 

32.  A  man  has  a  dollars,  h  dimes,  and  c  cents.  How  many 
cents  has  he  ? 

33.  A  room  is  x  yards  long,  and  y  yards  wide.  How  many 
square  yards  are  there  in  the  area  of  the  floor  ? 

34.  Find  the  area  of  the  floor  of  a  room  which  is  2  yards 
longer  and  3  yards  wider  than  the  one  mentioned  in  Ex.  33. 

35.  The  area  of  a  rectangular  field  equals  a  square  feet  and 
its  length  equals  b  feet.  Find  (a)  the  width  of  the  field. 
(b)   The  length  of  a  fence  surrounding  the  field. 

36.  The  sum  of  two  numbers  is  m-{-2n,  and  one  number 
equals  y.     Find  the  other  number. 

37.  By  what  must  7  be  multiplied  to  produce  a  product 
x-hy? 

38.  What  is  the  excess  of  m  +  ii  over  5-\-n? 

39.  What  is  the  cost  of  6  apples  at  x  cents  each  ? 

40.  What  is  the  cost  of  1  apple,  if  x  apples  cost  7  cents  ? 

41.  What  is  the  price  of  3  apples,  if  x  apples  cost  n  cents  ? 

42.  What  is  the  price  of  x  apples,  if  one  dozen  cost  n  cents  ? 

43.  A  man  bought  apples  at  the  rate  of  m  cents  per  dozen 
and  pears  at  the  rate  of  n  cents  per  dozen.  How  many  cents 
did  he  pay  for  x  apples  and  y  pears  ? 

44.  A  man  spent  x  dollars  in  buying  books  at  n  dollars  each. 
How  many  books  did  he  buy  ? 

45.  X  years  ago  a  man  was  20  years  old.  How  old  is  he 
now? 


LINEAR  EQUATIONS  AND  PROBLEMS  73 

46.  If  A  is  2  a?  4-  4  years  old,  how  many  more  years  must  lie 
live  to  be  80  years  old  ? 

47.  A  cistern  is  filled  by  a  pipe  in  a  minutes.  What  fraction 
of  the  cistern  will  be  filled  by  the  pipe  in  one  minute  ? 

48.  If  a  man  walks  4  miles  an  hour,  how  many  miles  will 
he  walk  in  n  hours  ? 

49.  If  a  man  walks  r  miles  an  hour,  how  many  miles  will 
he  walk  in  t  hours  ? 

50.  If  a  man  walks  20  miles  in  t  hours,  how  many  miles 
does  he  walk  each  hour  ? 

51.  If  a  man  walks  at  the  rate  of  r  miles  per  hour,  how 
many  hours  will  it  take  him  to  walk  20  miles  ? 

52.  How  many  miles  does  a  train  move  in  t  hours  at  the  rate 
of  X  miles  per  hour  ?     How  far  does  it  move  in  y  minutes  ? 

53.  A  man  traveled  a  miles  on  foot,  6  miles  by  boat,  and  the 
remainder  by  train.  How  many  miles  did  he  travel  by  train 
if  the  whole  journey  was  100  miles  ? 

54.  Find  5%  of  100  a.  56.    Find  «%  of  7. 

55.  Find  6%  of  x.  57.    Find  x%  of  a. 


97.  To  express  in  algebraic  symbols  the  sentence :  "a  exceeds 
h  by  as  much  as  h  exceeds  9,"  we  have  to  consider  that  in  this 
statement  "exceed"  means  minus  (— ),  and  "by  as  much  as" 
means  equals  (=).     Hence  we  have 

a  exceeds  h  by  as  much  as  c  exceeds  9. 

a       —      h  =  c       —       9. 

Similarly,  the  difference  of  the  squares  of  a  and  b  increased 

a^  -  62       4- 
by  80  equals  the  excess  of  a^  over  80. 

80      =  «»    -    80. 

Or,  (a2-62)_^80  =  a3-80. 


74  ELEMENTARY  ALGEBRA 

In  many  cases  it  is  possible  to  translate  a  sentence  word  by 
word  in  algebraic  symbols ;  in  other  cases  the  sentence  has  to 
be  changed  to  obtain  the  symbols. 

There  are  usually  several  different  ways  of  expressing  a  sym- 
bolical statement  in  words,  thus 

a—b=c  may  be  expressed  as  follows : 

The  difference  between  a  and  b  is  c. 

a  exceeds  h  by  c. 

a  is  greater  than  b  by  c. 

b  is  smaller  than  a  by  c. 

The  excess  of  a  over  b  is  c,  etc. 

EXERCISE   36 

Express  the  following  sentences  as  equations : 

1.  The  double  of  a  is  10. 

2.  One  third  of  b  is  17. 

3.  The  double  of  a  exceeds  one  third  of  b  by  c. 

4.  The  difference  of  a  and  b  increased  by  19  is  c. 

5.  Three  times  the  sum  of  a  and  b  exceeds  c  by  as  much  as 
c  exceeds  7. 

6.  Four  times  the  difference  of  x  and  y  increased  by  one 
fifth  of  c  is  equal  to  9  times  the  product  of  b  and  c. 

7.  The  product  of  the  sum  and  the  difference  of  a  and  b 
diminished  by  90  is  equal  to  the  sum  of  the  squares  of  a  and  b 
divided  by  7. 

8.  Twenty  subtracted  from  2  a  gives  the  same  result  as  7 
subtracted  from  a. 

9.  Nine  is  as  much  below  a  as  17  is  above  a. 

10.  a;  is  5%  of  720.  13.    20is9%ofa  +  6. 

11.  a:  is  6%  of  a.  14.    70isic%ofm. 

12.  50  isa;%  of  700. 


LINEAR   EQUATIONS  AND  PROBLEMS         '       75 

15.  If  A's  age  is  ic  +  20,  B's  age  is  4:X-\-12,  and  C's  age  is 
3  a;  + 10,  express  in  algebraic  symbols  the  following  statements : 

(a)  A  is  5  years  older  than  B. 

(b)  The  sum  of  A's  age  and  B's  age  is  60. 

(c)  A's  age  exceeds  B's  age  by  as  much  as  B's  age  exceeds 
C's  age. 

(d)  C  is  three  times  as  old  as  A. 

(e)  In  5  years  A  will  be  as  old  as  C  is  now. 

(/)  Five  years  ago  the  sum  of  B's  and  C's  ages  was  40. 

(g)  In  6  years  A  will  be  as  old  as  C  was  2  years  ago. 

(h)   X  years  ago  B's  age  was  20. 

(i)    In  X  years  the  sum  of  A's  and  B's  ages  will  be  70. 

(J)   One  half  of  A's  age  plus  one  third  of  B's  age  equals  40. 

16.  If  A,  B,  and  C  have  respectively  2  x,  Sx—  700,  and 
a; +  1200  dollars,  express  in  algebraic  symbols: 

(a)   A  has  5  dollars  more  than  B. 

(6)  If  A  gains  $  20  and  B  loses  ^  40,  they  have  equal  amounts. 

(c)  If  each  man  gains  $500,  the  sum  of  A's,  B's,  and  C's 
money  will  be  $12,000. 

(d)  A  and  B  together  have  $200  less  than  C. 

(e)  If  B  pays  to  C  $  100,  they  have  equal  amounts. 

(/)  If  A  pays  to  B  $50,  and  C  gains  $500,  then  A  and  C 
together  have  $  600  more  than  B. 

17.  A  sum  of  money  consists  of  x  dollars,  a  second  sum  of 
5x  —  30  dollars,  a  third  sum  of  2aj  +  1  dollars.  Express  as 
equations : 

(a)  5  %  of  the  first  sum  equals  $  90. 
(&)  a  %  of  the  second  sum  equals  $  20. 

(c)  x%  ot  the  first  sum  equals  6  %  of  the  third  sum. 

(d)  a  %  of  the  first  sum  exceeds  &  %  of  the  second  sum  by 
$900. 

(e)  4%  of  the  first  plus  5%  of  the  second  plus  6%  of  the 
third  sum  equals  $8000. 

(/)  ^%  oi  the  first  equals  one  tenth  of  the  third  sum. 


76  ELEMENTARY  ALGEBRA 

18.  Express  the  following  equations  in  words ;  using  for  the 
letters  x  and  y  the  words  "a  number"  and  "another  number." 

(a)  a;  +  12  =  22.  (e)  x-\- (Sx-7)  =  2x-{-ld. 

(6)  3^-12  =  17.  (f)x-y  =  2y-x. 

(c)  ^x-l  =  2x.  (g)  3a;-(a;-2)=122. 

(d)  1  +  ^  =  119.  (h)  ^^  +  y--3(x-y)  =  12(x  +  y). 

19.  If  A,  B,  and  C  have  respectively  x,  y,  and  z  dollars,  ex- 
press the  following  equations  as  verbal  statements : 

(a)  x  =  2y.  (f)z-y  =  200. 

(b)  x  =  y-^z.  ,  .   ^4-y  =  z 

(c)  3a;  =  400.  ^^^  4^3       * 

(d)  ji-^x  =  200.  (h)  a;  +  300  =  2  + 100. 

(e)  .05  a; +  .04  2/ =  212.  (i)  x  -  250  =  y -{- 100. 

PROBLEMS  LEADING  TO  SIMPLE   EQUATIONS 

98,  The  simplest  kind  of  problems  contain  only  one  unknown 
number.  In  order  to  solve  them,  denote  the  unknown  number  by 
X  {or  another  letter)  and  express  the  given  sentence  as  an  equation. 
The  solution  of  the  equation  gives  the  value  of  the  unknown  number. 

The  equation  can  frequently  be  written  by  translating  the 
sentence  word  by  word  into  algebraical  symbols;  in  fact  the 
'equation  is  the  sentence  written  in  algebraic  shorthand. 

Ex.  1.  Three  times  a  certain  number  exceeds  40  by  as  much 
as  40  exceeds  the  number.     Find  the  number. 

Let  X  =  the  number. 

Write  the  sentence  in  algebraic  symbols. 

Three  times  a  certain  no.  exceeds  40  by  as  much  as  40  exceeds  the  no. 
3x  X  -40=  40-  X 

Or,  3  a;  -  40  =  40  -  x. 

Transposing,  3  a;  +  x  =  40  +  40. 

Uniting,  4  a;  =  80v 

X  =  20,  the  required  number. 
Check.     8  ic  or  60  exceeds  40  by  20  ;  40  exceeds  20  by  20. 


LINEAR    EQUATIONS    AND    PROBLEMS  77 

Ex.  2.  In  15  years  A  will  be  three  times  as  old  as  he  was 
5  years  ago.     Find  A's  present  age. 

Let  a:  =  A's  present  age. 

The  verbal  statement  (1)  may  be  expressed  in  symbols  (2). 

(1)  In  16  years  A  will  be  three  times  as  old  as  he  was  5  years  ago. 

(2)  «     +  15        =        3         X  (x       -       5) 
Or,                                    a;  +  15  =  3(a;  -  5). 
Simplifying,                     a;  +  15  =  3  x  —  15. 
Transposing,                    fl!-33C  =  -15-16. 

Uniting,  -  2  x  =  -  30. 

Dividing,  '  x  =  16. 

Check,  In  15  years  A  will  be  30 ;  6  years  ago  he  was  10 ;  but 
30  =  3  X  10. 

Ex.  3.  To  a  quantity  of  water  contained  in  a  vessel,  56 
gallons  were  added,  and  there  was  then  in  the  vessel  8  times 
as  much  as  at  first.  How  many  gallons  did  the  vessel  contain 
at  first  ? 

Let  X  =  the  number  of  gallons  contained  in  the  vessel  at  first. 

The  verbal  statement  expressed  in  letters  gives  x  +  56  =  8  x; 

Transposing,  x  —  8  x  =  —  56. 

Uniting,  -  7  X  =  -  56. 

Dividing,  x  =  8,  the  required  number  of  gallons. 

Check.  If  56  gallons  are  added  to  8  gallons,  the  result  is  64  or  8  x  8 
gallons. 

Note.  The  student  should  note  that  x  stands  for  the  number  of 
gallons,  and  similarly  in  other  examples  for  number  of  dollars,  number 
of  yards,  etc. 

Ex.  4.     56  is  what  per  cent  of  120  ? 

Let  X  =  number  of  per  cent,  then  the  problem  expressed  in  symbols 
would  be 


66  =  -^.  120 
100 

or        fx  =  56. 

Dividing, 

X  =  46f . 

Hence 

66  is  46f  %  of  120. 

78  .    ELEMENTARY  ALGEBRA 

EXERCISE  37 

1.  What  is  the  number  which  when  subtracted  from  40  will 
give  the  same  result  as  when  added  to  14  ? 

2.  Find  the  number  whose  double  increased  by  4  equals  22. 

3.  What  number  added  to  three  times  itself  gives  a  sum  of 
44? 

4.  Find  the  number  whose  double  exceeds  7  by  5, 

5.  Four  times  a  certain  number  diminished  by  6  is  equal 
to  three  times  the  number  increased  by  2.     Find  the  number. 

6.  What  number  exceeds  6  by  as  much  as  three  times  the 
number  exceeds  24  ? 

7.  What  number  is  as  much  below  25  as  four  times  the 
number  is  above  30  ? 

8.  If  47  be  added  to  9  times  a  certain  number,  the  result 
will  be  11  times  the  excess  of  the  number  over  1.  Find  the 
number. 

9.  Five  times  a  certain  number  is  greater  by  4  than  six 
times  the  difference  of  the  number  and  2.     Find  the  number. 

10.  Thirty  years  hence  A  will  be  five  times  as  old  as  he  is 
now.     Find  his  present  age. 

11.  A  man  walks  a  certain  distance,  then  travels  three  times 
as  far  by  train,  and  then  travels  30  miles  by  boat.  If  the 
whole  journey  is  77  miles,  how  far  does  he  walk  ? 

12.  Twenty-eight  years  hence  a  man  will  be  twice  as  old  as 
he  will  be  two  years  hence.     How  old  is  he  now  ? 

13.  Fifteen  years  hence  a  man  will  be  twice  as  old  as  he 
was  5  years  ago.     How  old  is  he  now? 

14.  To  each  of  the  numbers  1, 13,  and  5,  an  unknown  number 
is  added.  If  the  product  of  the  first  two  sums  is  equal  to  the 
square  of  the  last  sum,  what  is  the  number  ? 


LINEAR  EQUATIONS  AND  PROBLEMS  79 

15.  A  cistern  is  filled  by  a  pipe,  and  the  quantity  let  in  after 
8  minutes  is  45  gallons  more  than  the  quantity  let  in  after  5 
minutes.     Find  the  number  of  gallons  let  in  per  minute. 

16.  A  train  moving  at  a  uniform  rate  runs  in  5  hours  41 
miles  more  than  in  3  hours.  How  many  miles  per  hour  does 
it  run  ? 

17.  A  man  is  33  years  old,  and  his  son  is  12  years  old. 
How  many  years  ago  was  the  father  four  times  as  old  as 
the  son? 

18.  A  man  is  36  years  old,  and  his  son  is  11  years  old.  How 
many  years  hence  will  the  father  be  twice  as  old  as  the  son  ? 

19.  60  is  5  %  of  what  number  ? 

20.  12  is  what  per  cent  of  22  ? 

21.  A  has  $20,  and  B  has  $30.  How  many  dollars  must  B 
give  to  A  to  make  A's  money  equal  to  4  times  B's  money  ? 

22.  John  and  Henry  have  the  same  number  of  marbles.  If 
John  buys  20  marbles  more  and  Henry  loses  10,  then  John 
will  have  four  times  as  many  as  Henry.  How  many  has 
each? 

23.  A  man  bought  two  houses  for  the  same  price.  He  sold 
one  at  a  profit  of  $  3000,  and  the  other  at  a  loss  of  $  1500, 
receiving  twice  as  much  for  the  first  as  for  the  last.  How 
much  did  he  pay  for  the  houses  ? 

24.  A  man  wished  to  purchase  a  farm  containing  a  certain 
number  of  acres.  He  found  one  farm  which  contained  20  acres 
too  many,  and  another  which  lacked  15  acres  of  the  required 
number.  If  the  first  farm  contained  twice  as  many  acres  as 
the  second  one,  how  many  acres  did  he  wish  to  buy  ? 

25.  If  the  sixth  part  of  a  number  be  added  to  18,  the  result 
is  the  same  as  if  three  quarters  of  the  number  were  subtracted 
from  29.     Find  the  number. 


80  ELEMENTABY  ALGEBRA 

99.  If  a  problem  contains  two  unknown  quantities,  two  verbal 
statements  must  be  given.  In  the  simpler  examples  these  two 
statements  are  given  directly,  while  in  the  more  complex  prob- 
lems they  are  only  implied.  We  denote  one  of  the  unknown 
numbers  (usually  the  smaller  one)  by  Xj  and  use  one  of  the 
given  verbal  statements  to  express  the  other  unknown  number 
in  terms  of  x.  The  other  verbal  statement,  written  in  algebraic 
symbols,  is  the  equation,  which  gives  the  value  of  x. 

Ex.  1.  One  number  exceeds  another  by  8,  and  their  sum  is 
14.     Find  the  numbers. 

The  problem  consists  of  two  statements : 
I.    One  number  exceeds  the  other  one  by  8. 

II.    The  sum  of  the  two  numbers  is  14. 

Either  statement  may  be  used  to  express  one  unknown 
number  in  terms  of  the  other,  although  in  general  the  simpler 
one  should  be  selected. 

If  we  select  the  first  one,  and 

Let  X  ~  the  smaller  number, 

Then  x  -f  8  =  the  greater  number. 

The  second  statement  written  in  algebraic  symbols  produces 
the  equation  x  +  {x^S)=lL 

Simplifying,  a;  -f-  a;  ■+•  8  =  14. 

Transposing,  a;  +  ic  =  14  —  8 

Uniting,  2x  —  Q. 

Dividing,  a;  =  3,  the  smaller  number. 

a;  +  8  =  11,  the  greater  number.  ■ 

Another  method  for  solving  this  problem  is  to  express  one  unknown 
quantity  in  terms  of  the  other  by  means  of  statement  II ;  viz.  the  sum  of 
the  two  numbers  is  14. 

Let  X  =  the  smaller  number. 

Then,  14  —  a;  =  the  larger  number. 

Statement  I  expressed  in  symbols  is  (14  —  x)  —  x=8,  which  leads  of 
course  to  the  same  answer  as  the  first  method. 


LINEAR   EQUATIONS  AND  PROBLEMS  81 

Ex.  2.  A  has  three  times  as  many  marbles  as  B.  If  A 
gives  25  marbles  to  B,  B  will  have  twice  as  many  as  A. 

The  two  statements  are : 

I.  A  has  three  times  as  many  marbles  as  B. 

II.   If  A  gives  B  25  marbles,  B  will  have  twice  as  many  as  A. 

Use  the  simpler  statement,  viz,  I,  to  express  one  unknown  quantity  in 
terms  of  the  other. 

Let  X  =  B's  number  of  marbles. 

Then,  Sx  =  A's  number  of  marbles. 

To  express  statement  II  in  algebraic  symbols,  consider  that  by  Xhe 
exchange  A  will  lose,  and  B  will  gain. 

Hence,      x  +  25  =  B's  number  of  marblies  after  the  exchange. 
3  a;  —  25  =  A's  number  of  marbles  after  the  exchange. 

Therefore,  jc  +  25  =  2  (3  re  -  25).  (Statement  II) 

Simplifying,  x  +  25  =  6  x  —  60. 

Transposing,  x  —  6  x  =  —  25  —  50. 

Uniting,  —  5x  =  -75. 

Dividing,  x  =  15,  B's  number  of  marbles. 

3  X  =  45,  A's  number  of  marbles. 

Check.      45  -  25  =  20,  15  +  25  =  40,  but  40  =  2  x  20. 

100.  The  numbers  which  appear  in  the  equation  should  always 
be  expressed  in  the  same  denomination.  Never  add  the  number 
of  dollars  to  the  number  of  cents,  the  number  of  yards  to  their 
price,  etc. 

Ex.  3.  Eleven  coins,  consisting  of  half  dollars  and  dimes, 
have  a  value  of  $  3.10.     How  many  are  there  of  each  ? 

The  two  statements  are  : 
I.   The  number  of  coins  is  11. 

II.  The  value  of  the  half  dollars  and  dimes  is  $8.10. 

Let,  X  =  the  number  of  dimes,  then,  from  I, 

11  —  X  =  the  number  of  half  dollars. 
o 


82  ELEMENTARY  ALGEBRA 

Selecting  the  cent  as  the  denomination  (in  order  to  avoid  fractions)  we 

express  the  statement  II  in  algebraic  symbols. 

50(ll-a:)  +10x  =  310. 
Simplifying,  560  -  50  a;  +  10  a;  =  310. 

Transposing,  —  50  a;  +  10  a;  =  —  550  +  310. 

Uniting,  _  40  re  =  -  240. 

Dividing,  ■  x  =  6,  the  number  of  dimes. 

11  —  a;  =  5,  the  number  of  half  dollars. 

Check.  6  dimes  =  60  cents,  5  half  dollars  =  250  cents,  their  sum  is 
$3.10. 

EXERCISE  38 

1.  Two  numbers  differ  by  33,  and  the  greater  is  four  times 
the  smaller.     Find  the  numbers. 

2.  Find  two  numbers  whose  sum  is  72  and  the  greater  of 
which  equals  five  times  the  smaller. 

3.  The  difference  between  two  numbers  is  8,  and  if  16  be 
added  to  the  greater,  the  result  will  be  three  times  the  smaller. 
Find  the  numbers. 

4.  The  difference  between  two  numbers  is  2,  and  the  dif- 
ference between  their  squares  is  16.     Find  the  numbers. 

5.  The  sum  of  two  numbers  is  47,  and  their  difference  is  9. 
Find  the  numbers. 

6.  Divide  20  into  two  parts,  one  of  which  increased  by  14 
shall  be  equal  to  the  other  increased  by  10. 

7.  One  number  is  5  less  than  three  times  another  number. 
If  the  second  number  is  subtracted  from  five  times  the  first 
number,  the  result  is  25.     What  are  the  numbers  ? 

8.  Divide  22  into  two  parts  such  that  one  part  multiplied 
by  5  is  equal  to  the  other  part  diminished  by  2. 

9.  Find  two  consecutive  numbers  whose  sum  is  equal  to 
243. 


LINEAR  EQUATIONS  AND  PROBLEMS  83 

10.  Find  two  consecutive  numbers,  the  difference  of  whose 
squares  is  equal  to  27. 

11.  A's  age  is  three  times  B's,  and  in  10  years  A's  age  will 
be  twice  B's.     Find  their  ages. 

12.  A  and  B  divide  a  sum  of  money.  A  receives  $5  more 
than  B,  and  five  times  B's  money  diminished  by  three  times  A's 
money  equals  $  25.     How  much  does  each  receive  ? 

13.  The  length  of  a  rectangular  field  is  three  tintes  its  width, 
and  a  fence  surrounding  the  field  is  248  yards.  Find  the  length 
and  width  of  the  field. 

14.  A  is  27  years  older  than  B,  and  B's  age  is  as  much  below 
20  as  A's  age  is  above  33.     What  are  their  ages  ? 

15.  Two  vessels  contain  together  7  pints.  If  the  smaller 
contained  2  pints  more,  it  would  contain  half  as  much  as  the 
larger  one.     How  many  pints  are  there  in  each  vessel  ? 

16.  On  December  21,  the  night  in  St.  Petersburg  lasts  13 
hours  longer  than  the  day.    How  many  hours  does  the  day  last  ? 

17.  The  difference  of  two  numbers  is  8,  and  their  sum  is  five 
times  the  smaller.     Find  the  two  numbers. 

18.  A  man  built  a  house  costing  three  times  as  much  as  the 
lot.  If  the  house  cost  $  8000  more  than  the  lot,  what  was  the 
price  of  each  ? 

19.  A  line  45  inches  long  is  divided  into  two  parts.  Twice 
the  larger  part  exceeds  three  times  the  smaller  part  by  30 
inches.     How  many  inches  are  there  in  each  part  ? 

20.  A  has  $  16  less  than  B.  If  B  gives  $  20  to  A,  A  will 
have  five  times  as  much  as  B.     How  many  dollars  has  each  ? 

21.  A  commenced  business  with  three  times  as  much  capital 
as  B.  During  the  first  year  A  lost  $  600,  B  gained  $  200,  and 
A  had  then  only  twice  as  much  as  B.  How  much  capital  had 
each  at  first  ? 


84  ELEMENTARY  ALGEBRA 

22.  A  man  has  f  G.25  in  half  dollars  and  quarters.  He  has 
three  times  as  many  quarters  as  half  dollars.  How  many  half 
dollars  and  quarters  has  he  ?     (  Ex.  3,  §  100.) 

23.  The  sum  of  $3.50  is  made  up  of  19  coins,  which  are 
either  half  dollars  or  dimes.     How  many  are  there  of  each? 

24.  John  and  Henry  together  have  50  marbles.  If  John 
had  6  marbles  more,  he  would  have  three  times  as  many  as 
Henry.     How  many  has  each? 

25.  John  and  Henry  together  have  60^.  If  John  gives 
Henry  6^  he  will  have  4  times  as  much  as  Henry.  How  many 
cents  has  each  ? 

26.  Henry  bought  13  apples,  some  at  the  rate  of  4^  per 
apple,  the  rest  at  3^  per  apple.  How  many  did  he  buy  of 
each  kind  if  he  paid  45;^  in  all? 

27.  A  and  B  together  buy  200  lbs.  of  sugar.  A  takes  113 
lbs.  and  B  takes  the  remainder.  If  A  uses  3^  lbs.  per  day  and 
B  uses  2^  lbs.  per  day,  after  how  many  days  will  they  have 
equal  quantities  of  sugar? 


101.  If  a  problem  contains  three  unknown  quantities,  three 
verbal  statements  must  be  given.  One  of  the  unknown  numbers 
is  denoted  by  x,  and  the  other  two  are  expressed  in  terms  of  x 
by  means  of  two  of  the  verbal  statements.  The  third  verbal 
statement  produces  the  equation.  If  four  or  more  unknown 
quantities  occur  in  the  problem,  the  method  is  similar. 

If  it  should  be  difficult  to  express  the  selected  verbal  state- 
ment directly  in  algebraical  symbols,  try  to  obtain  it  by  a  series 
of  successive  steps. 

Ex.  1.  A,  B,  and  C  together  have  $  80,  and  B  has  three 
times  as  much  as  A.  If  A  and  B  each  gave  f  5  to  C,  then 
three  times  the  sum  of  A's  and  B's  money  would  exceed  C's 
money  by  as  much  as  A  had  originally. 


LINEAR  EQUATIONS  AND  PROBLEMS  85 

The  three  statements  are : 

I.   A,  B,  and  C  together  have  $80. 
II.   B  has  three  times  as  much  as  A. 

III.    If  A  and  B  each  gave  $5  to  C,  then  three  times  the  sum  of  A's 
and  B's  money  would  exceed  C's  money  by  as  much  as  A  had  originally. 
Let  X  =  the  number  of  dollars  A  has. 

According  to  II,  Sx  =  the  number  of  dollars  B  has, 

and  according  to  I,      80  —  4a;  =  the  number  of  dollars  C  has. 

To  express  statement  III  by  algebraical  symbols,  let  us  consider  first 
the  words  "if  A  and  B  each  gave  $5  to  C." 

x  —  6  =  number  of  dollars  A  had  after  giving  $  6. 
3  X  —  6  =  number  of  dollars  B  had  after  giving  $  5. 
90  —  4  03  =  number  of  dollars  C  had  after  receiving  $  10. 
Expressing  in  symbols : 

Three  times  the  sum  of  A's  and  B's  money  exceeds  C's  money  by  A's 
3  X  (a:-5  +  3x-5)  -        (90-4a;)    =    x. 

original  amount. 

The  solution  gives      x  =  S,  number  of  dollars  A  had. 

3x  =  24,  number  of  dollars  B  had. 

80  —  4  a:  =  48,  number  of  dollars  C  had. 

Check.  If  A  and  B  each  gave  $5  to  C,  they  would  have  3,  19,  and  58 
respectively.     3  (3  +  19)  or  66  exceeds  58  by  8. 

Ex.  2.  A  man  spent  $1185  in  buying  horses,  cowSj  and 
sheep,  each  horse  costing  $90,  each  cow  $35,  and  each  sheep 
$15.  The  number  of  cows  exceeded  the  number  of  horses  by 
4,  and  the  number  of  sheep  was  twice  as  large  as  the  number 
of  horses  and  cows  together.  How  many  animals  of  each  kind 
did  he  buy  ? 

The  three  statements  are  : 

I.   The  total  cost  equals  $1185. 

II.   The  number  of  cows  exceeds  the  number  of  horses  by  4. 

III.  The  number  of  sheep  is  equal  to  twice  the  number  of  horses  and 
cows  together. 


86  ELEMENTARY  ALGEBRA 

Let  X  r=  the  number  of  horses, 

then,  according  to  II,      x  +  4  =  the  number  of  cows, 
and,  according  to  III, 

2  (2  X  +  4)  or  4  X  +  8  =  the  number  of  sheep. 
Therefore,  90  x  =  the  number  of  dollars  spent  for  horses. 

35  (x  +  4)  =  the  number  of  dollars  spent  for  cows, 
and,  15  (4  X  +  8)  =  the  number  of  dollars  spent  for  sheep. 

Hence  statement  I  may  be  written, 

90  X  +  35  (X  +  4)  +  15  (4  X  4- 8)  =  1185. 
Simplifying,  90  x  +  35  x  +  140  +  60  x  +  120  =  1185. 
Transposing,  90  x  +  35  x  +  60  x  =  -  140  -  120  +  1185. 

Uniting,  185x  =  925. 

Dividing,  x  =  5,  number  of  horses. 

.  X  +  4  =  9,  number  of  cows. 

4  X  +  8  =  28,  number  of  sheep. 

Check.  5  horses,  9  cows,  and  28  sheep  would  cost  5  x  90+  9  x  35  + 
28  X  15  or  450  +  315  +  420  =  1185  ;  9-5  =  4;  28  =  2  (9  +  6). 

EXERCISE   39 

1.  Find  three  numbers  sucli  that  the  second  is  three  times 
the  first,  the  third  is  four  times  the  first,  and  the  difference 
between  the  third  and  the  second  is  five. 

2.  Find  three  numbers  such  that  the  sum  of  the  first  two  is 
14,  the  third  is  twice  the  first,  and  the  third  exceeds  the  second 
by  4. 

3.  Find  three  numbers  such  that  the  second  exceeds  the 
first  by  3,  the  third  exceeds  the  first  by  10,  and  the  third  is 
twice  the  first. 

4.  Find  three  numbers  such  that  the  sum  of  the  first  and 
second  is  5,  the  sum  of  the  first  and  last  is  6,  and  the  last  is 
twice  the  first. 

6.  Find  three  numbers  such  that  the  sum  of  the  first  two  is 
8,  the  second  exceeds  the  last  by  one,  and  the  square  of  the  last 
exceeds  the  square  of  the  first  by  7, 


LINEAR  EQUATIONS  AND  PROBLEMS  87 

6.  The  difference  between  two  numbers  is  4 ;  a  third  number 
is  5  less  than  the  sum  of  the  first  two ;  and  the  sum  of  the  first 
and  third  numbers  is  14.     What  are  the  numbers  ?  . 

7.  Divide  20  into  three  parts  such  that  the  second  part  is 
twice  the  first,  and  the  first  part  exceeds  the  last  by  4. 

8.  A  is  three  times  as  old  as  B,  and  C  is  five  years  younger 
than  A.  Three  years  ago  the  sum  of  their  ages  was  56  years. 
What  are  their  ages  ? 

9.  A  is  five  years  older  than  B,  and  three  years  younger 
than  C.  Seventeen  years  ago  C  was  twice  as  old  as  B.  Find 
the  age  of  each. 

10.  A  man  has  5  sons  each  three  years  older  than  the  next 
younger.  The  age  of  the  eldest  three  years  hence  will  be  three 
times  the  present  age  of  the  youngest.     Find  the  age  of  each. 

11.  Divide  50  into  three  parts  such  that  the  first  part  is  8 
more  than  the  second,  and  the  third  increased  by  34  is  twice  as 
large  as  the  sum  of  the  first  and  second  parts. 

12.  A  is  five  years  older  than  B,  and  C  is  three  times  as  old 
as  B  was  five  years  ago.  In  five  years  C's  age  will  be  10  times 
the  difference  between  A's  and  B's  ages.    Find  the  age  of  each. 

13.  The  three  angles  of  any  triangle  are  together  equal  to 
180°.  If  the  second  angle  of  a  triangle  is  10°  larger  than  the 
first,  and  the  third  is  twice  the  sum  of  the  first  and  second, 
what  are  the  three  angles  ? 

14.  There  are  420  sheep  in  three  flocks.  The  second  con- 
tains twenty  sheep  more  than  the  first,  and  the  third  twice  as 
many  as  the  first.     How  many  sheep  are  there  in  each  flock  ? 

15.  There  are  500  sheep  in  three  flocks.  The  second  con- 
tains twice  as  many  as  the  first,  and  if  10  be  taken  from  the 
third  and  added  to  the  first,  the  third  will  contain  as  many  as 
the  first  and  second  together.  How  many  sheep  are  there  in 
each  flock  ? 

16.  Find  three  consecutive  numbers  whose  sum  is  126. 


88  ELEMENTARY  ALGEBRA 

17.  Find  four  consecutive  numbers  such  that  the  last  is 
twice  the  first. 

18.  Find  three  consecutive  numbers  such  that  the  difference 
of  the  squares  of  the  third  and  first  is  20. 

19.  A,  B,  and  C  divide  a  certain  sum  of  money.  A  receives 
$50  more  than  B.  A  and  C  together  receive  $300,  and  B  has 
$  150  less  than  C.     How  much  does  each  receive  ? 

20.  Three  boys,  A,  B,  and  C,  divide  a  certain  number  of 
marbles,  so  that  A  and  B  together  receive  22,  A  and  C  25,  and 
B  and  C  27.     How  many  marbles  does  each  receive  ? 

21.  A,  B,  and  C  have  together  240  sheep.  B  has  twenty 
more  than  A,  and  C  has  as  many  as  A  and  B  together.  How 
many  sheep  has  each  ? 

22.  B  has  twice  as  many  acres  as  A,  and  G  has  three  times 
as  many  acres  as  B.  If  A  and  C  together  have  1400  acres, 
how  many  acres  has  each  ? 

23.  A  has  one  third  as  much  money  as  B.  C  has  $  25  more 
than  A,  and  $  15  more  than  B.     How  much  has  each  ? 

24.  In  a  room  there  were  twice  as  many  women  as  children, 
and  three  more  men  than  children.  The  number  of  men  and 
women  together  was  9.    How  many  children  were  there  present  ? 

25.  A  horse,  carriage,  and  harness  cost  $300.  The  horse 
costs  $10  less  than  the  carriage  and  $70  more  than  the  har- 
ness ;  and  the  carriage  and  harness  together  cost  $  180.  Find 
the  cost  of  the  horse. 

26.  In  3  classes  there  are  120  pupils.  The  second  class  con- 
tains 10  more  than  the  first,  and  the  first  and  third  together 
have  80  pupils.     How  many  pupils  are  there  in  each  class  ? 

27.  The  sum  of  the  four  angles  of  any  quadrilateral  is  360°. 
If  the  second  angle  is  twice  as  large  as  the  first,  the  third  twice 
as  large  as  the  second,  and  the  fourth  30°  larger  than  the  third, 
find  the  value  of  each  angle. 


LINEAR   EQUATIONS  AND  PROBLEMS 


89 


28.  A  has  as  many  pennies  as  B  has  dollars,  C  has  as  many 
five-dollar  bills  as  B  has  dollars,  and  together  they  have  $  72.12. 
How  much  has  each  ? 

29.  Three  farms  contain  together  1280  acres.  The  first  con- 
tains 200  acres  more  than  f  of  the  second,  and  the  third  10 
acres  less  than  |  of  the  second.  How  many  acres  does  each 
farm  contain  ? 

30.  The  capacity  of  the  first  of  three  barrels  is  |  of  that  of 
the  second,  and  the  capacity  of  the  second  ^^  that  of  the  third. 
If  the  contents  of  the  third  barrel  be  poured  into  the  empty 
first  barrel,  there  will  be  10  gallons  left.  How  many  gallons 
does  each  barrel  hold  ? 

31.  The  surface  of  the  earth  consists  of  5  zones,  the  torrid, 
two  temperate,  and  two  frigid  zones.  Each  temperate  zone  is 
Jl  of  the  torrid,  each  frigid  is  ^-^  of  each  temperate,  and  the 
surface  of  the  earth  is  approximately  200,000,000  square  miles. 
How  many  square  miles  does  each  zone  contain  ? 


102.  Arrangement  of  Problems.  If  the  example  contains 
quantities  of  3  or  4  different  kinds,  such  as  length,  width,  and 
area,  or  time,  speed,  and  distance,  it  is  frequently  advantageous 
to  arrange  the  quantities  in  a  systematic  manner. 

E.g.  A  and  B  start  at  the  same  hour  from  two  towns  27  miles 
apart,  B  walks  at  the  rate  of  4  miles  per  hour,  but  stops  2 
hours  on  the  way,  and  A  walks  at  the  rate  of  3  miles  per  hour 
without  stopping.  After  how  many  hours  will  they  meet  and 
how  many  miles  does  A  walk? 


Time 
(In  hours) 

Katb 

(miles  per  hour) 

Distance 

(miles) 

A  .     .     .     . 

X 

3 

3x 

B  .     .     .     . 

x-2 

4 

4(x-2) 

90  ELEMENTARY  ALGEBRA 

Explanation.     First  fill  in  all  the  numbers  given  directly,  i.e.  3  and  4. 

Let  X  =  number  of  hours  A  walks,  then  x  —  2  =  number  of  hours  B 
walks.  Since  in  uniform  motion  the  distance  is  always  the  product  of 
rate  and  time,  we  obtain  3  x  and  4  (x  —  2)  for  the  last  column.  But  the 
statement  "A  and  B  walk  from  two  towns  27  miles  apart  until  they 
meet ' '  means  the  sum  of  the  distances  walked  by  A  and  B  equals  27  miles. 

Hence  3x  +  4  (x  -  2)  =  27. 

Simplifying,      3a:  +  4x-8  =  27. 
Uniting,  7  x  =  35. 

Dividing,  x  =  5,  number  of  hours. 

Sx=  15,  number  of  miles  A  walks. 

This  is,  of  course,  not  a  new  method  for  solving  problems, 
but  simply  a  convenient  mode  of  arranging  the  solutions  of 
some  examples,  which,  however,  may  be  solved  without  this 
arrangement. 

Whenever  various  denominations  occur  repeatedly  in  the 
same  connection,  or  when  similar  operations  have  to  be  per- 
formed repeatedly  with  several  quantities,  this  arrangement 
will  be  found  advantageous.  Examples  in  which  one  quantity 
is  found  by  multiplying  the  numerical  values  of  two  or  more 
quantities  belong  to  this  group.  In  the  following  list  the 
numerical  values  of  the  last  column  are  equal  to  the  products 
of  the  numerical  values  of  the  first  two  columns,  provided  care 
is  taken  in  regard  to  the  denomination. 

Length  width  and  area  of  rectangle 

Kate  of  speed  time  distance  covered 

Number    of   per-  number  of  dollars  number  of  dollars  all 

sons  each  paid  paid 

Number  of  yards  price  per  yard  total  cost 

Number  of  coins  value  of  each  coin  total  value  of  coins 

Principal  rate  per  cent  interest 

Ex.  1.  The  length  of  a  rectangular  field  is  twice  its  width. 
If  the  length  were  increased  by  30  yards,  and  the  width  decreased 
by  10  yards,  the  area  would  be  100  square  yards  less.  Find 
the  dimensions  of  the  field. 


LINEAR  EQUATIONS  AND  PROBLEMS 


91 


Length 
(yards) 

Width 
(yards) 

Akea 

(square  yards) 

First  field  .... 

2x 

X 

2  0^2 

Second  field    .     .     . 

2a;+30 

x-10 

(2  X  +  30)  (X  -  10) 

The  area  would  be  decreased  by  100  square  yards  gives 

(2  X  +  30)  (X  -  10)  =  2  ic2  -  100. 
Simplify,  2  a;2  +  lo  x  -  300  =  2x2  -  100. 

Cancel  2  x2  and  transpose,  10  x  =  200. 

x  =  20. 
2  X  =  40. 
The  field  is  40  yards  long  and  20  yards  wide. 

Check.    The  original  field  has  an  area  40  x  20  =  800,  the  second  field 
70x10  or  700.    But  700  =  800  -  100. 

Ex.  2.   A  certain  sum  invested  at  5%  brings  the  same  in- 
terest as  a  sum  $200  larger  at  4%.     What  is  the  capital? 


Principal 
(No.  of  dollars) 

Rate  % 

Interest 
(No.  of  dollars) 

X 

.D5 

.05  X 

x  +  200 

.04 

.04  (x  +  200) 

Therefore  .  05  x  =  .  04  (x  +  200) . 

Simplify,  .05  x  =  .04  x  +  8. 

Transposing  and  uniting,    .01  x  =  8. 

Multiplying,  x  =  800  ;  $  800  =  required  sum. 

Check.  $800  X  .05  =  $40  ;  $1000  x  .04  =  $40. 


EXERCISE  40 

1.  A  rectangular  field,  is  20  yards,  and  another  25  yards  wide. 
The  second  is  10  yards  longer  than  the  first,  and  the  sum  of 
their  areas  is  equal  to  1600  square  yards.  Find  the  dimensions 
of  each. 


92  ELEMENTARY    ALGEBBA 

2o  A  rectangular  field  is  14  yards  longer  than  it  is  wide. 
If  its  length  were  increased  by  10  yards,  and  its  width  de- 
creased by  4  yards,  the  area  would  remain  the  same.  Find 
the  dimensions  of  the  field. 

3.  A  rectangular  field  is  twice  as  long  as  it  is  wide.  If  it 
were  50  feet  shorter  and  20  feet  wider,  it  would  contain  2000 
square  feet  less.     Find  the  dimensions  of  the  field. 

4.  A  certain  sum  invested  at  4^  brings  the  same  interest  as 
a  sum  $300  larger  invested  at  3%.     Pind  the  first  sum. 

5.  A  sum  invested  at  5%,  and  a  second  sum,  twice  as  large_, 
invested  at  4%,  together  bring  $52  interest.  What  are  the 
two  sums  ? 

6.  An  investment  of  $  2500  brings  a  yearly  interest  of  $114. 
A  part  of  the  capital  is  invested  at  4%,  and  the  remainder  at 
5%.     How  many  dollars  are  invested  at  4%  ? 

7.  A  bought  12  oranges  for  a  certain  sum.  If  each  orange 
had  cost  one  cent  more,  he  would  have  received  10  oranges  for 
the  same  money.     What  was  the  price  of  each  orange  ? 

8.  Six  persons  bought  an  automobile,  but  as  two  of  them 
were  unable  to  pay  their  share,  each  of  the  others  had  to  pay 
$40  more.  Find  the  share  of  each,  and  the  cost  of  the 
automobile. 

9.  Ten  yards  of  silk  and  20  yards  of  cloth  cost  together 
$35.  If  the  silk  cost  three  times  as  much  per  yard  as  the 
cloth,  how  much  did  each  cost  per  yard  ? 

10.  Twenty  yards  of  silk  and  30  yards  of  cloth  cost  together 
$  85.  If  the  silk  cost  50  ^  more  per  yard  than  the  cloth,  what 
was  the  price  of  each  per  yard? 

11.  A  man  bought  7  lbs.  of  coffee  for  $1.79.  For  a  part  he 
paid  24  ^  per  lb.  and  for  the  rest  he  paid  35  ^  per  lb.  How 
many  pounds  of  each  kind  did  he  buy  ? 

12.  Sixteen  persons  subscribed  $138.  Six  of  them  paid 
equal  amounts,  and  the  remaining  ones  paid  each  one  dollar 
more.     Find  the  share  of  each  man. 


LINEAR  EQUATIONS  AND  PROBLEMS  93 

13.  Twenty  men  subscribed  equal  amounts  to  raise  a  certain 
sum  of  money,  but  four  men  failed  to  pay  their  shares,  and  in 
order  to  raise  the  required  sum  each  of  the  remaining  men  had 
to  pay  one  dollar  more.      How  much  did  each  man  subscribe  ? 

14.  A  cistern  is  filled  in  a  certain  time  by  a  pipe  which  lets 
in  20  gallons  per  minute.  Another  pipe  letting  in  25  gallons 
per  minute  fills  the  cistern  in  one  minute  less.  In  how  many 
minutes  does  the  first  pipe  fill  the  cistern  ? 

15.  A  cistern  is  filled  in  a  certain  time  by  a  pipe  letting  in 
21  gallons  per  minute.  If  another  pipe,  which  lets  in  14  gal- 
lons per  minute,  is  opened  3  minutes  longer  than  the  first,  6 
gallons  less  than  in  the  first  case  will  be  poured  in.  In  how 
many  minutes  does  the  first  pipe  fill  the  cistern  ? 

16.  A  sets  out  walking  at  the  rate  of  3  miles  per  hour,  and 
three  hours  later  B  follows  on  horseback  traveling  at  the  rate 
of  6  miles  per  hour.  After  how  many  hours  will  B  overtake 
A,  and  how  far  will  each  then  have  traveled? 

17.  A  and  B  set  out  walking  at  the  same  time  in  the  same 
direction,  but  A  has  a  start  of  3  miles.  If  A  walks  at  the  rate 
of  2i  miles  per  hour,  and  B  at  the  rate  of  3  miles  per  hour,  how 
far  must  B  walk  before  he  overtakes  A  ? 

18.  A  sets  out  walking  at  the  rate  of  3  miles  per  hour,  and 
one  hour  later  B  starts  from  the  same  point  traveling  by  coach 
in  the  opposite  direction  at  the  rate  of  6  miles  per  hour.  After 
how  many  hours  will  they  be  27  miles  apart  ? 

19.  A  and  B  start  walking  at  the  same  hour  from  two  towns 
17|-  miles  apart,  and  walk,  toward  each  other.  If  A  walks  at 
the  rate  of  3  miles  per  hour,  and  B  at  the  rate  of  4  miles  per 
hour,  after  how  many  hours  do  they  meet  and  how  many  miles 
does  A  walk  ? 

20.  The  distance  from  New  York  to  Albany  is  142  miles. 
If  a  train  starts  at  Albany  and  travels  toward  New  York  at  the 
rate  of  40  miles  per  hour  without  stopping,  and  another  train 


94  ELEMENTARY  ALGEBRA 

starts  at  the  same  time  from  'New  York  traveling  at  the  rate 
of  42  miles  an  hour,  how  many  miles  from  New  York  will  they 
meet? 

21.  Two  men  start  at  12  o'clock  from  two  towns  17  miles 
apart,  and  travel  toward  each  other.  One  walks  at  the  rate  of 
3  miles  per  hour,  but  rests  one  hour  on  the  way;  the  other 
travels  at  the  rate  of  4  miles  per  hour  and  rests  3  hours.  At 
what  hour  do  they  meet  ? 

22.  A  and  B  start  from  two  towns  20  miles  apart  and  travel 
toward  each  other.  A  starts  at  1  p.m.,  B  starts  at  2  p.m.,  and 
they  meet  at  5  p.m.  If  B  travels  one  mile  per  hour  faster  than 
A,  find  the  number  of  miles  each  travels  per  hour. 

23.  A  picture  which  is  2  inches  longer  than  wide  is  sur- 
rounded by  a  frame  1  inch  wide.  If  the  area  of  the  frame  is 
40  square  inches,  what  are  the  dimensions  of  the  picture  ? 

MISCELLANEOUS  PROBLEMS 

24.  The  formula  which  transforms  Fahrenheit  readings  of  a 
thermometer  into  Centigrade  readings  is  C  =|(F  — 32). 

If  C  =  40°,  find  the  value  of  F. 

25.  Change  the  following  readings  to  Fahrenheit  readings : 
(a)  0°  C,  (6)  100°  C,  (c)  50°  C,  (d)  -12°  C. 

26.  At  what  temperature  do  the  Centigrade  scale  and  Fah- 
renheit scale  indicate  equal  numbers  ? 

27.  Th^  formula  for  the  distance  which  a  falling  body 
passes  over  in  t  seconds  is  S  =  ^gf.     (Ex.  7,  p.  16.) 

If  S  =  240  ft.  and  ^  =  4  seconds,  find  the  value  of  g. 

28.  The  formula  for  compound  interest  is 

(For  the  meaning  of  the  letters  see  Ex.  4,  p.  16.) 

Find  the  principal  that  will  bring  $  662  interest  in  two  years 
at  10  %  compound  interest. 


LINEAR    EQUATIONS    AND    PROBLEMS  95 

29.  A  number  increased  by  7  gives  the  same  result  as  the 
number  multiplied  by  7.     What  is  the  number  ? 

30.  If  a  number  be  added  to  3,  the  sum  multiplied  by  3,  the 
product  diminished  by  20,  the  difference  multiplied  by  6,  and 
the  product  diminished  by  55j  the  result  will  be  5.  Find  the 
number. 

31.  A  has  as  many  dollars  as  B  has  cents.  If  A  should  give 
B  ^  6.93,  B  would  have  as  many  dollars  as  A  has  cents.  How 
much  money  has  each  ? 

32.  A  man  made  as  much  money  as  he  had  and  $100.  He 
made  as  much  money  as  he  then  had  and  $200;  again  he 
made  as  much  as  he  then  had  and  $  300,  and  found  that  he 
had  finally  $  3100.     How  many  dollars  had  he  at  first  ? 

33.  A  man  met  some  beggars,  and  after  giving  each  4  ^  had 
9^  left.  He  found  that  he  lacked  7^  to  be  able  to  give  each 
beggar  6^.     How  many  beggars  were  there  ? 

34.  A  mason  working  8  hours  a  day,  in  the  course  of  a  week, 
builds  a  number  of  cubic  meters  which  exceeds  43  as  much  as 
43  exceeds  the  number  of  cubic  meters  which  he  would  build 
working  7^  hours  a  day.  How  many  cubic  meters  does  he 
build  per  hour  ? 

35.  A  has  f  6  more  than  B  and  gives  to  B  as  much  as  B 
has.  Then  B  gives  to  A  as  much  as  A  then  has,  and  once  more 
A  gives  to  B  as  much  as  B  then  has ;  and  finds  that  A  has 
now  as  much  as  B.     How  many  dollars  has  each  at  first  ? 

36.  A  boy  has  the  same  number  of  sisters  as  brothers,  while 
his  sister  has  1^  times  as  many  brothers  as  sisters.  How 
many  sons  and  daughters  are  there  in  the  family  ? 


CHAPTER   VI 
FACTORING 

103.  An  expression  is  rational  with  respect  to  a  letter,  if, 
after  simplifying,  it  contains  no  indicated  root  of  this  let- 
ter; irrational,  if  it  does  contain  some  indicated  root  of  this 
letter. 

cfi h  V6  is  rational  with  respect  to  a,  and  irrational  with  respect 

to&.      "^      , 

104.  An  expression  is  integral  with  respect  to  a  letter,  if 
this  letter  does  not  occur  in  any  denominator. 

—  +  a6  +  62  is  integral  with  respect  to  a,  but  fractional  with  respect 
h 

to  6. 

105.  An  expression  is  integral  and  rational,  if  it  is  integral 
and  rational  with  respect  to  all  letters  contained  in  it;  as, 

a2  +  2  a&  +  4  c^. 

106.  The  factors  of  an  algebraic  expression  are  the  quantities 
which  multiplied  together  will  give  the  expression. 

In  the  present  chapter  only  integral  and  rational  expressions 
are  considered  factors. 

Although  Va^  -  h^  x  Va^  -h^^o?  -  h\  we  shall  not,  at  this 
stage  of  the  work,  consider  ■\/a?  —  6^  a  factor  of  o?  —  51 

107.  A  factor  is  said  to  be  prime,  if  it  contains  no  other 
factors  (except  itself  and  unity) ;  otherwise  it  is  composite. 

The  prime  factors  of  10  a^b  are  2,  5,  a,  a,  a,  h. 

96 


FACTORING  97 

108.  Factoring  is  the  process  of  separating  an  expression  into 
its  factors.  An  expression  is  factored  if  written  in  the  form 
of  a  product. 

(x2  -  4  tK  4-  3)  is  factored  if  written  in  the  form  (x  —  3)  (a;  —  1).  It 
would  not  be  factored  if  written  x{x  —  4)  +  3,  for  this  result  is  a  sum, 
and  not  a  product. 

109.  The  factors  of  a  monomial  can  be  obtained  by 
inspection. 

The  prime  factors  of  12  x^y^  are  3,  2,  2,  x,  x,  x,  y,  y. 

110.  Since  factoring  is  the  inverse  of  multiplication,  it  fol- 
lows that  every  method  of  multiplication  will  produce  a 
method  of  factoring. 

E.g.  since  (a  +  6)  (a  —  6)  =  a^  —  h"^,  it  follows  that  a^  —  6^  can  be 
factored,  or  that  a^  —  6^  =  (a  +  6)  (a  —  6). 

111.  Factoring  examples  may  be  checked  by  multiplication 
or  by  numerical  substitution. 


TYPE  I.    POLYNOMIALS  ALL  OF  WHOSE   TERMS 
CONTAIN  A   COMMON  FACTOR 

mx  +  my-\-  mz  =  m{x  -\-y  +  z).     (§  55.) 

112.   Ex.  1.   Factor  6  x^  -  9  a^2/«  +  12  xy\ 

The  greatest  factor  common  to  all  terms  is  3  xy'^.     Divide 

6  xV  -9  a;2y8  +  12  xy*'  by  3  xy"^, 

and  the  quotient  is  2  oj^  —  3  :Ky  +  4  yK 
But,  dividend  =  divisor  x  quotient. 

Hence  6  7?y^  -  9  ^s^/S  4. 12  ojj/*  =  3  a^(2  a;2  -  3  xy  +  4  y2). 

Ex.  2.   Factor 

14  a462c2  -  21  a264c2  +  7  a'^V^c^  =  7  a^HK"^  a^  _  3  52  +  1). 


98  ELEMENTARY  ALGEBEA 

EXERCISE  41 
Factor  the  following  expressions : 

1.  15abx-9b^x.  9.   4  a^ft  -  5  aV  +  6  ac?«. 

2.  9a^-6a\  '       10.    17  mW-51mV+85  mn. 

3.  16a^-4ic2.  11.   15  a'b'x- 9  b^y-\- 12  b\ 

4.  Uacd-7cd-\-21d'd\  12.    9xyz'-6xYz^+3xYz^. 

5.  3a»-6a2  +  9a.  13.  Ux^y^-21a^fz+4:9  a^yh\ 

6.  Sp*y-\-2py-6pY.  14.  12mV-18mV-24mW. 

7.  5a;2^2_i5^2/  +  20a;?/«.        .  15.  llpV-33J^Y^-llpV• 
8.    Im  —  lmn  —  lp.  IQ.  x"^ -\- x? —  x^ -\-x. 

17.  39a-^6V-26a^6V  +  13a^6V. 

18.  51  mY  —  34  my  + 12  my. 

19.  2a;^-4a^2/  +  6a^/  +  8?/3. 

20.  4  a;*2/2  -  28  a^2/3  ^  49  a;2^4  _  43  ^y^ 

21.  a;(a  +  ft)  +  2/(«  +  &). 

22.  3o?(m  +  n)-2f{m  +  n). 

23.  6  a262(j>  +  g)  -  4  ab\p  +  g)  -  (p  +  g). 

24.  ^x^{x-y)-l  z\x-y),         27.  a;2(a;  -  3)  -  3  a;(a;  -  3). 

25.  4  ic'*  -  12  x'^+i  -  6  0?'^+^  28.  3x\x  +  9) -{x  +  9), 

26.  6  a2«5"  -  3  a"62«.  29.  am  +  bm  +  an  +  bn. 

TYPE  XL    QUADRATIC   TRINOMIALS  OF   THE   FORM 

x^  +  px  -\-  q. 

113.  In  multiplying  two  binomials  containing  a  common 
term,  e.g.  (a;  —  3)  and  {x-\-5),  we  had  to  add  —3  and  5  to  ob- 
tain the  coefficient  of  x,  and  to  multiply  —3  and  5  to  obtain  the 
term  which  does  not  contain  a;  or  (a;  —  3)(a?  +  5)  =  ar^  +  2  a;  — 15. 


FACTORING  '      99 

In  factoring  oi? -[-2x—15  we  have,  obviously,  to  find  two 
numbers  whose  product  is  —15  and  whose  sum  is  +2. 

Or,  in  general,  in  factoring  a  trinomial  of  the  form  x^+px-^-q, 
we  have  to  find  two  numbers  m  and  n  whose  sum  is  p,  and 
whose  product  is  g;  and  if  such  numbers  can  be  found,  the 
factored  expression  is  {x  -\-m)(x-{-  n), 

Ex.  1.   Factor  x^-4.x-77. 

We  may  consider  —77  as  the  product  of  —1  •  77,  or  — 7  •  11,  or  — 11  •  7, 
or  -77  .  1,  but  of  these  only  —11  and  7  have  a  sum  equal  to  -4. 
Hence  a;2 -4x  -  77  =(5c  -  ll)(a;  +  7). 

Since  a  number  can  be  represented  in  an  infinite  number  of 
ways  as  the  sum  of  two  numbers,  but  only  in  a  limited  number 
of  ways  as  a  product  of  two  numbers,  it  is  advisable  to  consider 
the  factors  of  q  first.  If  q  is  positive,  the  two  numbers  have 
both  the  same  sign  as  p.  If  q  is  negative,  the  two  numbers 
have  opposite  signs,  and  the  greater  one  has  the  same  sign  as  p. 

Not  every  trinomial  of  this  type,  however,  can  be  factored. 

Ex.  2.    Factor  a^-lla-^ 30. 

The  two  numbers  whose  product  is  30  and  whose  sum  is  —11  are  —5 
and  —6. 

Therefore  a"^  -  n  a  +  SO  =(a  -  5)(a  -  6). 

Check.     Ifa  =  l,  a2-ll  a+30=20,  and  (a-5)(a-6)  =  -4  . -5  =  20. 

Ex.  3.   Factor  a^-^  10  ax -11  a\ 

The  numbers  whose  product  is  —  11  a^  and  whose  sum  is  10  a  are  11  a 
and  —a. 

Hence  x2  +  10  ax  -  11  a^  =  (x  +  11  a)  (x  -  a). 

Ex.  4.   Factor  ic«  -  7  a^?/^  +  12  y^ 

The  two  numbers  whose  product  is  equal  to  12  y^  and  whose  sum  equals 
-7y3  are  -iy^  and  -3^3.     Hence  x6-7xV+12  2/«  =  (x3-3  y3)(x»-4y8). 

Ex.  5.   Factor  1  -  3  a  - 10  a^. 

This  expression  is  a  special  form  of  the  general  type,  obtained  by  let- 
ting X  =  1. 

Hence  1  -  3  a  -  10  a2  =  (1  -  5  a)Cl  +  2  a). 


100  ELEMENTARY   ALGEBRA 

114.   In  solving  any  factoring  example,  the  student  should  first 
determine  whether  all  terms  contain  a  common  monomial  factor. 

EXERCISE  42 

Factor  the  following  expressions ;  * 

1.  a?-5x-\-Q.  24.  m2  4-28mn  +  187n«. 

2.  a;2  +  5a;  +  6.  25.  ory^ +  2  ah-Zb. 

3.  a2-3a  +  2.  26.  x^yh'^  -  I'd  xyz -\- AS>. 

4.  a^  +  la  +  12.  27.  a^-4.a'-21. 

5.  a;2_43._2i.  28.  aj«  +  17a^  +  60. 

6.  m2  +  4m-21.  29.  a^  - 11  a^d^  +  24  6*. 

7.  ^2  4.5g_i4.  30.  a^6^-13a2W-30c*. 

8.  2/^-7?/-18.  31.  9m4-m2  +  20. 

9.  /-82/  +  15.  32.  a2  +  262_3a6. 

10.  0^-5  a; -14.  33.  7-8  m  +  ml 

11.  x'^  +  2x-[-l.  34.  3m  — 4  + m^. 

12.  m^- 14m +  33.  35.  l-7m  +  12m2. 

13.  a^  — 3  a;  — 4.  36.  m^  — 25. 

14.  2/'-38^  +  37.  37.  a^-bx'  +  Qx. 

15.  2/2_362/-37.  38.  3a2-36a  +  33. 

16.  m2-19m  +  48.  39.  3  m^- 15 m2-18 m. 

17.  h^-Uh-bl.  40.  bf~imy-20y\ 

18.  a;2-32a;  +  175.  41.  Q>  ah^ -^  a%  -  ^  d'h^ 

19.  a^-8a;  +  16.  42.  6  a^  +  2  a^-36aj. 

20.  V  - 17  aa;  + 30  a^.  43.  98  a:?/^  -  28  0;^  +  2  a^. 

21.  a2-12a6-1362.  44.  a^-7aj'«  +  12. 

22.  m^  + 15  mil  -  34  7i2.  45.  7^^+^— 9  ffi'^hi +  20  m^n\ 

23.  p" -Ipqr -{-12  qh^  46.  {a  +  hf  -1  {a-\-h)  -1%. 

47.  (a  4- &)'- 12  (a +  6)2 +  20  (a +  6). 

48.  o?-Zab. 


FACTORING  101 

TYPE   III.     QUADRATIC  TRINOMIALS  OF  THE  FORM 
px^  +  qx  +  r. 

115.   According  to  §  66 j 

20  x^  is  the  product  of  4  a;  and  5  x. 
—  6  is  the  product  of  +  3  and  —  2. 
+  7  ic  is  the  sum  of  the  cross  products. 

Hence  in  factoring  6x'^  —  lSx-\-5,  we  have  to  find  two  bino- 
mials whose  corresponding  terms  are  similar,  such  that 

The  first  two  terms  are  factors  of  6  a^. 

The  last  two  terms  are  factors  of  5, 

and  the  sum  of  the  cross  products  equals  — 13  x. 

By  actual  trial  we  find  which  of  the  factors  of  6^^  and  o 
give  the  correct  sum  of  cross  products. 

If  we  consider  that  the  factors  of  +  5  must  have  like  signs, 
and  that  they  m.ust  be  negative,  as  —13x  is  negative,  all  possi- 
ble combinations  are  contained  in  the  following  : 

6x  —  l  6x  —  5  3a;— 1  3a;— -5 

X      X      ^      ^/ 


5  x-1  2a;-5  2a;- 


—  31  a;  -  11  a;  -  17  a;  -  lo  a; 

Evidently  the  last  combination  is  the  correct  one,  or 
6x'-lSx  +  5  =  {Sx-5){2x-l). 

116.  In  actual  work  it  is  not  always  necessary  to  write  down 
all  possible  combinations,  and  after  a  little  practice  the  student 
should  be  able  to  find  the  proper  factors  of  simple  trinomials 
at  the  first  trial.  The  work  may  be  shortened  by  the  following 
considerations : 

1.  i/"  p  is  positive,  only  positive  factor's  of  p  need  be  considered. 

2.  If  p  and  r  a7'e  positive,  the  second  terms  of  the  factors  have 
the  same  sign  as  q. 


102    •  ELEMENTARY  ALGEBRA 

S.  If  ip  is  positive,  and  r  is  negative,  then  the  second  terms  of 
the  factors  have  opposite  signs. 

If  a  combination  should  give  a  sum  of  cross  products,  which  has  the 
same  absolute  value  as  the  term  qx,  but  the  opposite  sign,  exchange  the 
signs  of  the  second  terms  of  the  factors. 

4.  If  px^  -{-qx-\-r  does  not  contain  any  monomial  factor,  none 
of  the  binomial  factors  can  contain  a  monomial  factor. 

Ex.  1.     Factor  3  a^  -  83  a;  +  54. 

The  factors  of  the  first  term  consist  of  one  pair  only,  viz,  3  x  and  x, 
and  the  signs  of  the  second  terms  are  minus.  54  may  be  considered  the 
product  of  the  following  combinations  of  numbers :  1  x  54,  2  x  27,  3  x  18, 
6  X  9,  9  X  6,  18  X  3,  27  X  2,  54  X  1.  Since  the  first  term  of  the  first  factor 
(3  x)  contains  a  3,  we  have  to  reject  every  combination  of  factors  of  54, 
whose  first  factor  contains  a  3.  Hence  only  1  x  54  and  2  x  27  need  be 
considered. 


Sx-2 
-83x 


Therefore  Sx^ -8Sx  +  54  =  (Sx-2)(x- 27). 

Ex.  2.     Factor  9x^-^20x-21. 

9  aj2  =  9  X  •  a;  or  3  a;  •  3  X,  but  the  second  combination  has  to  be  rejected,  as 
21  contains  a  3,  and  as  consequently  one  of  the  resulting  binomial  factors 
would  contain  the  monomial  factor  3.     Hence  9x-x  has  to  be  selected. 

The  last  term  21  may  be  factored  as  follows :  1  x  21,  3  x  7,  7  x  3,  and 
21  X  1.    According  to  (4),  only  1  x  21  and  7x3  need  to  be  tried. 


9a;  +  l  9a;  +  7 


rx  -f-  1  vjo  -f-  I 

X  X 

K-21  a; -3 


-188ic  -20x 

The  second  combination  produces  the  absolute  value  of  the  middle  term 
but  the  wrong  sign,  hence  the  factors  are 

(9x-7)(a;  +  3). 


FACTOBING  103 

117.  The  type  pa? -{- qx -\- r  is  the  most  important  of  the 
trinomial  types,  since  all  others  (II,  IV)  are  special  cases  of 
it.  In  all  examples  of  this  type,  the  expressions  should  be 
arranged  according  to  the  ascending  or  the  descending  powers 
of  some  letter,  and  the  monomial  factors  should  be  removed. 

EXERCISE  43 

Factor  the  following  expressions  : 

1.  ^:^-x-2.  21.  2a2-3a6-262. 

2.  5a2-9a-2.  22.  Aa^-a-U. 

3.  3a^-10a;  +  3.  23.  60  a"  -  59  ab  -  20  b^. 

4.  4:X^  +  7x-2.  24.  12a;*-23a^  +  10. 

5.  3a2-5a  +  2.  25.  8  a«-38  a^  +  SS. 

6.  2x'-9x  +  A.  26.  2-5a'  +  3a\ 

7.  9x2-26a;-3.  27.  3-a;-2a^. 

8.  4ar^-8aj  +  3.  28.  6-x-2i^. 

9.  4:X^-llx-S.  29.  12-2x'-5x. 

10.  6b^  +  b -12.  30.  12 -a^- a;. 

11.  4a^-5aj-6.  31.  -5x-x'-\-6. 

12.  6a2-19a  +  10.  32.  4. x^ +10 xy -\- 4c y^ 

13.  9m2-17m-2.  33.  S  x^y^ +22  xy^  -  6  y^, 

14.  5x2  +  26a;  +  5.  34.  24  «« +  42  a^d -  45  ad^. 

15.  6a2_17a  +  12.  35.  30 aj^^/ +  95 a^i/^ - 35 a;^/^ 

16.  4a2-4a6-3&2.  se.  3  aj"+24.  5  a;«+i-f2a;". 

17.  4  +  13a;H-3a^.  37.  2  (a -{■  bf -\- 11  (a -{- b)  +  5. 

18.  6a;2-7an/-3/.  38.  4.(x+yy-S(x  +  y) +3. 

19.  2a2  +  13a6  +  66-.  39.  3a^"-a;"-2. 

20.  15a2  — 77a  +  10.  40.  aa?  +  (a-\-b)x  +  b. 


104  ELEMENTARY  ALGEBRA 

TYPE  IV.    THE   SQUARE   OF   A  BINOMIAL 

118.  Expressions  of  this  form  are  special  cases  of  the  pre- 
ceding type,  and  may  be  factored  according  to  the  method  used 
for  that  type.  In  most  cases,  however,  it  is  more  convenient 
to  factor  them  according  to  §  65. 

x^  +  2xij-\-i/  =  {x  +  yy. 
af  —  2  xy  -\- y^  —  (x  —  yy. 

A  trinomial  belongs  to  this  type,  i.e.  it  is  a  perfect  square, 
when  two  of  its  terms  are  perfect  squares,  and  the  remaining 
term  is  equal  to  twice  the  product  of  the  square  roots  of  these 
terms. 

The  student  should  note  that  a  term,  in  order  to  be  a  perfect 
square,  must  have  a  positive  sign. 

16 a;2  —  24 xy  +  9 y"^  is  a  perfect  square,  for  2Vl6x2  x  \/9p  =  24 xy. 
Evidently  16x^  -  2ixy  -i-  9y'^  =  {^x  -  S yy. 

To  factor  a  triyiomial  which  is  a  perfect  square,  connect  the 
square  roots  of  the  terms  which  are  squares  by  the  sign  of  the  re- 
maining term,  and  indicate  the  square  of  the  resulting  binomial. 

EXERCISE  44 

Determine  whether  the  following  expressions  are  perfect 
squares  or  not,  and  factor  whenever  possible: 

1.  a'  +  2ab-\-b\  7.  p^  +  ^p-\-16. 

2.  a^-2ab-b\  8.  l-4.m-\-4.m\ 

3.  c?-\-2cd-d?,  9.  9  +  6p+jp^ 

4.  ar2  +  4a;H-4.  10.  4.-6p-\-f, 

5.  a^-6a:  +  9.  11.  a2-14aH-49. 

6.  a;2-2a;  +  4.  ^2.  ce^ -f  18 a  +  81. 


FACTORING  105 

13.  a2  +  6a-9.  25.  -a?-\-2ah-h\ 

14.  a2-6a6  +  96^  26.  1^^ y"^  +  x" - 2Q xy, 

15.  4ft2_l2a64-962.  27.  a2  +  2a5. 

16.  9mV  +  42mn  +  49.  28.  +  12  a;'*  +  36  ar'"  + 1. 

17.  4a^/-20a;2/;2+2522.  29.  2mV-m^-?i«. 

18.  4a;^-15a;2^V4-9?/V.  30,  36  a.-^  +  6  a^  +  54  a;. 

19.  36a^-60«2?,2_^25  6^  31.  a2--2a"6'"  + &'" 

20.  225a;«-30a;3  +  l.  32.  22«-6.2'^  +  9. 

21.  4a2-8a6  +  462.  33.  98  a^y  -  56  .<?/ +  8  a:^^,*. 

22.  m^  +  2m-  +  m.  34.  25  a^y  -  101  a;/ +  4  2/^ 

23.  m^-12m26^-36  6^  35.  4.  a' -  52  a^ -{- lU  a\ 

24.  2a2  4-12a6-18  62.  36.  (a  +  6)^  -  24  (a  +  6)  + 144. 

37.    {a-by-2x(a-b)  +  x\ 

TYPE  V.    THE  DIFFERENCE   OF   TWO   SQUARES 

119.   According  to  §  65, 

a'-b'  =  (a  +  b)(a-b)y 

i.e.  tJie  difference  of  the  squares  of  two  numbers  is  equal  to  the 
product  of  the  sum  and  the  difference  of  the  two  numbers. 

Ex.  1.      4a^/-9z«  =  (2aj32/*  +  3  2;«)(2aj32/4-3«3). 

Ex.  2.    16a«-64  6^«  =  16(a8-4&io) 

=  16(a<  +  2  6*)(a*-2  6«). 

Ex.  3.  a*  -  6^  =  (a^  +  b%a^  -  b") 

=  {a'-i.b')(a  +  b){a-by 
Note,     a^  _|.  52  jg  prime. 


106  ELEMENTABY  ALGEBRA 

EXERCISE  45 

Resolve  into  prime  factors : 

1.  m^-n\  16.  4a^-9a6^l 

2.  p'-q^.  17.  lx'^~ly^\ 

3.  a--^.  18.  16aj9-196». 

4.  16-51  19.  9ai<'-81. 

5.  4a2-l.  20.  100-900ai««. 

6.  l-^ba'hK  21.  a^-al 

7.  a"62_25cl  22.  144  a^-a^ 

9.    49a^6*-16c^  24.    12a»-3  6V. 

10.  36  a8- 25  6^8  35     a^m_i^ 

11.  169  c''- 121  (^«e«.  26.  64  a^"*  -  9  fe^"*. 

12.  225  a^  -  144  a62.  27.  32--22-. 

13.  a^-b^  28.  25  a2+- -  225  a^d^. 

14.  a4_54  29.  10000-1. 

15.  a»-6».  30.  99.91. 

120.   One  or  both  terms  are  squares  of  polynomials. 

Ex.  1.    Factor  a^  -  (c  +  d)l 

a2  _  (c  +  d)2  =  (a  +  c  +  d)  (a  -  c  -d). 
Ex.  2.    Eesolve  into  prime  factors  and  simplify 
(4  a +3  5)2- (2  a-5  6)2. 

(4a+3  6)2-(2a-6  6)2=:[(4a+3  5)  +  (2a-5  5)][(4a  +  3  6)-(2a-5  6)] 
=  [4a  +  3  6  +  2a-5  6][4a+3  6-2a  +  5&] 
=  (6a-2  6)(2a+8  6) 
=2(3a-6)  .  2(a+4  6) 
=4(3a-5)(a+4  5). 


FACTORING  107 

EXERCISE  46 

Resolve  into  prime  factors  : 

1.  (a-{-by-(f.  7.  (a^-2ay-(a-by. 

2.  (x-yy-z\  8.  (4:a-hby-(3x-yy. 

3.  (a  +  by-dc".  9.  l-(3a-5by. 

4.  a'-(b  +  cy.  10.  3-3(a  +  &)'. 

5.  4a2_(6_c)2.  11.  (a4-&  +  c)2-(a;  +  2/-2)2. 

6.  (a-\-3by-16(f.  12.  (a4-2  6+3c-f-c?)=^-(e+/)'. 
Eesolve  into  factors  and  simplify : 

13.  (a-\-by-a\  17.  (3  a +  5)'- (2  a- 1)^. 

14.  a" -(a -by,  18.  (3  a^  -  a)^  -  (a  +  6)1 

15.  (a  +  2&)2-9a2.  19.  (4a  +  6  6)2- (a-7  6/. 

16.  25a2-(2a-6  6)2.  20.  (a  +  6 -f- c)^ -  (a  +  6)1 

TYPE  VI.     THE   SUM  OR  DIFFERENCE  OF   TWO   CUBES 

jr^+/^,'  and  x^  —  ^. 

121.   According  to  §  79 : 

3?^f  =  {x-\-y){x'-xy  +  f) 
a^-y^  =  (x-  y)(x^  +  xy  +  y^), 
Ex.  1.    Factor  8  0^-1. 

8x6-1  =  (2x2)8-   (1)3 

=  (2x2-l)(4x*  +  2x2  +  l). 

Ex.  2.   Factor  8  a^  +  27  6^ 

8  rt9  -{-  27  &6  =  (2  a3)3  ^.  (3  52)3 

=  (2  a3  +  3  62)(4  a6  -  6  a^h^  +  9  6*). 

After  a  little  practice  the  student  may  omit  the  intermediate  step  and 
write  the  factors  at  once. 

Ex.  3.   125aV-343c» 

=  (5  a'b*  -  7  c3)(25  a^6«  +  35  a26V  +  49  c«). 


108  ELEMENTARY  ALGEBRA 

EXERCISE  47 
Factor  the  following : 

1.  a^-b\  10.   125  a' +  51  19.  3aff-81x^\ 

2.  a'  +  b\  11.    8a^-27b\  20.  ic^^  - 1331. 

3.  a^-1.  12.   216a3  +  125  6l  21.  x'-2'^. 

4.  a^  +  1.  13.    1000 -a;27.  22.  1001. 

5.  1+a^  14.   216  a^-b'c".  23.  1,000,001. 

6.  S-a\  15.    x^-27afy^  24.  1,000,027. 

7.  Sa^  +  l.  16.    2  0^  +  54  2/3.  25.  64,001. 

8.  27  6^-1.  17.    a''-\-b'\  26.  64,000,001. 

9.  64a3-6l  18.    a^  +  27a5^.  27.  999,973. 


122.  In  factoring  a^  —  b^,  the  expression  may  be  considered 
either  the  difference  of  two  squares  or  the  difference  of  two 
cubes,  producing  respectively  the  following  results : 

a^-b^=(a^-^¥)(a^-b^) 

a^  -b'=  (a2  -  b')(a'  +  a'b''  +  b'). 

The  factors  of  the  first  result  can  easily  be  factored  again, 
while  it  is  difficult  to  factor  the  last  factor  of  the  second  re- 
sult. Hence  the  prime  factors  of  examples  of  this  type  can  be 
obtained  most  readily  by  considering  them  the  difference  of 
two  squares. 

Ex.   a6_64  =  (a3  +  8)(a3-8) 

=  (a  +  2)(a'  -  2  a  +  4)(a  -  2)(a'  +  2  a  +  4). 

EXERCISE  48 
MISCELLANEOUS  EXAMPLES 
Besolve  into  prime  factors : 

1.  l-x\  3.    aW-729.  5.    1-mVl 

2.  m^-n^  4.    a^'^  -  b^  6.    a'^-b'^. 


FACTORING  109 

7.  «^-4«2  +  3.  17.   a«62-26a36*-27&«. 

8.  x'-2ar-3.  18.    a^  -  5  a^ -\- 4.. 

9.  3a*-3a2-36.  19.   aji^-aj. 

10.  4m«-32m^-36m*.  20.  a^-5a«+4. 

11.  a4-5a262_36  6*.  21.  729  a^d^  _  a^ft*. 

12.  a^  +  4:a3-5a.  22.  (a +  &)*-!. 

13.  a^-«6«.  "  23.  (a  +  6/-l. 

14.  3i)«-39p*  +  108i>«.  24.  (aj  +  2/)'--l. 

15.  18x*-74a^/  +  8  2/*.  25.  3«"-2^". 

16.  x^-7a^-S. 

TYPE  VII.    GROUPING  TERMS 

123.  By  the  introduction  of  parentheses,  polynomials  can 
frequently  be  transformed  into  bi-  and  trinomials,  which 
may  be  factored  according  to  types  I- VI. 

A.  After  groupi7ig  the  terms,  we  find  that  the  new  terms  contain 
a  common  factor, 

Ex.  1.   Factor  ax -{- hx -\- ay  -\-  by. 

ax+  bx -^  ay  -\- by  =  x{a  +  6)  -\-y{a  +  6) 
=  (a  +  6)(x  +  y). 

Ex.  2;  Factor  a^  -  5  a;^  —  a;  +  5. 

a;8-5a:2-a;  +  6  =  x2(x  -  5)-  (x  -  5) 
=  (x-b-)(x^-\) 
=  (x-5)(x  +  l)(x-l). 

EXERCISE  49 

1.  xm  ■\- ym -\- xn -\' yn.  4.    2  am  +  2  op  — 3  6m  — 3  6p. 

2.  »^  +  a;?/ +  aa;  +  ai/-  5.    6  am —  3  6m  — 6  an +  3  ftn. 

3.  2aa;-36a;  +  2ay-3  6y.        6.    2  y'' -  y^ -\-4.y-2. 


110  ELEMENTARY  ALGEBRA 

7.  p^<f  —  p-q^  —  pq -\- 1.  14.    a^  —  a  —  a^xy -\- xy. 

8.  af  +  mxy  —  4  a;i/  —  4  my^.        15.  p'^  +pq'  —p^q  —  g^. 

9.  6  x^-\-3xy  —  2  ax  — ay.  16.    1  — a;  — a^  +  ar^. 

10.  c^dHe'c^'-cy^-ey^.  17.  p^-5p'-\-2p-10. 

11.  3iB^  — 7fl^  +  3a7  — 7.  18.  a^  —  a^x  —  ap  +  px. 

12.  6a;*-13a^-12a;+26.  19.  m«- 13  m*-7  m^ +  91. 

13.  x^  —  x^  —  X -\- 1.  20.  aa;+6aj+a2/4-62/+^^  +  ^2;. 

It  is  sometimes  necessary  to  change  the  order  of  the  terms  of 
the  given  expression  before  the  method  can  be  applied. 

21.  ax  +  by  —  ay—bx. 

22.  aj3-21  +  3a;-7a^. 

23.  aV-3&y  +  36V-ay. 


B.   By  grouping,  the  expression  becomes  the  difference  of  two 
squares. 

Ex.  1.   Factor  a^-6ab  +  b^-ie c\ 

a2  _  6  a6  +  9  62  _  16  c2  =  (a2  _  6  a&  +  9  62)  _  16  c2 
=  (a-3  6)2-(4c)2 
=  (a  -  3  6  +  4  c)  (a  -  3  6  -  4  c). 

Ex.2.    Factor  9a^-/-4;22_|_4  2^^^ 

gx^-y2-4:z^  +  4:yz  =9 x^ -(y^  -  4yz  +  4: z^) 
=  (Sxy-(y-2zr- 
=  {Sx  +  y-2z)(iSx-y-^2z), 

Ex.3.   YsictoT4:a^-b^-{-9x^-4:y^-12ax-{-4:by. 

Arranging  the  terms, 

4  a2  -  62  +  9  x2  -  4  2/2  -  12  ajc  +  4  6y 

=  4  a2  -  12  aic  +  9  x2  -  62  +  4  6y  -  4?/2 
=  (4  a2  -  12  ax  +  9  x2)  _  (62  _  4  6y  +  4  y^) 
=  (2«-3x)2-(6-2y)2 
=  (2a-3x+6-2t/)C2a-3x-6-f2  2^)'. 


FACTORING  111 

EXERCISE  50 

1.  a?-2db-{-h^-l.  7.  2^0" -2xy -x" -y\ 

2.  a?-4.ah-{-4.h^--(?.  8.  a?J\-h^-{.2ab-Qc'-\-2xy-y\ 

3.  x^-Qxy  +  ^y'^-2ba\  9.  l&x'' -4:0? -A.ab -h\ 

4.  l-a2  +  2a6-6l  lO.  3 -3a2  +  6a6-36l 

5.  16m2-4a^-4a;2/-2/^.  H.  a  -  a^- Ga^ft  _  9a62. 

6.  16-a2-624_2a6.  12.  a^  -  10  a; -1- 25  -  121  c«. 

13.  a2  +  12a  +  36-a^  +  4iC2/-42/^- 

14.  a"  -  a?  -\-h''  -y^  -2  ah  -2xy. 

15.  a^  -  2  a^ft^.^  6* -25x^-40x2?/' -16?/*. 


(7.   By  grouping,  the  expression  becomes  a  trinomial  of  the  form 
of  pa^  +  qx-\-r  (or  its  special  cases  II  and  IV). 

Ex.  1.     Factor  3a.'2-6a;2/  +  3/-10a;  +  102/  +  3. 
3a;2  _  6x2/ +  3?/2  -  lOa;  +  lOy  +  3  =  3(a:2  -  2x?/ +  ?/2)- 10(a;  -  y)4- 3 

=  3(x-y)2-10(x-y)  +  3 
=.[3(x-2/)-l][(x-2/0-3] 
=  [3x-32/-l][x-2/-3]. 

Similar,  although  a  type  of  VII A  is  the  following  example : 
Ex.  2.     4a^-12a^4-9/-2a;  +  32/. 

4  x2  -  12  xy  +  9  ?/2  -  2  X  +  3  ?/  =  (2  X  -  3  ?/)2  -  (2  X  -  3y) 
=  (2x-32/)(2x-3t/-l). 

EXERCISE  61 

1.  a2_^2a  +  l  +  a&  +  ^  3.    a2_4«54.452_3^^g5_j.2. 

2.  a^-2ah  +  h^-a  +  h.  4.    ic^ _ ^,2 _j_ 2 a; - 2 2/. 

5.  a.'2-3a;2/  +  2y2_2a;4-42/. 

6.  0^^-6x2/4-92/^- TxH- 21 2/ -f  12. 

7.  4a24-8a6  +  462_5a-56  +  l. 

(Eor  additional  methods  of  factoring,  see  Appendix  II  and 
Chapter  XVI.) 


112  ELEMENTARY  ALGEBRA 

SUMMARY  OF  FACTORING 

I.   First  find  monomial  factors  common  to  all  terms. 
II.  Binomials  are  factored  by  means  of  the  formulae 

a^-b^  =  (a-hb){a-b). 

a'Jrb'=(a-{-b)(a'-ab-{-b'). 

III.  Trinomials  are  factored  by  the  method  of  cross  products, 
although  frequently  the  particular  cases  II  and  IV  are  more  con- 
venient. 

IV.  Polynomials  are  reduced  to  the  preceding  cases  by  grouping 
terms. 

EXERCISE  52 
MISCELLANEOUS  EXAMPLES 
Factor  the  following  expressions : 

1.  a2-6l  14.    17  a2- 25  a -18. 

2.  a'  +  b^-2ab.  15.    20 a^  -  220  a  +  605. 

3.  a2  +  2a6.  16.    a^-729a. 

4.  a^-Sab-\-2bK  17.    7a»-7. 

5.  Sa^-10ab  +  Sb\  18.    7 a^ -\- 77 a^b -  S4: ab^ 

6.  6ab-~Sa'-3bK  19.    a" -~  c' +  b^ -2ab. 

7.  a^-a'b.  20.    2  a^  +  16  a6  - 130  ft^. 

8.  a^-ab\  21.    a^- 1-26-62. 

9.  2xy  —  x^  —  y^.  22.   x^  —  ax  —  bx -{- ab. 

10.  a^-16a^-{-55x.  23.  2a^  +  x'-Sa^. 

11.  a^-16a.  24.  2  —  x-x^. 

12.  8x^-650^  +  8x2.  26.  (a  +  64-c)2-l. 
18.  Sax  +  2bx-{-3ay-\-2by.  26.  l-(a-f-6)3. 


FACTOBING  113 

27.  o^-y\  42.    a.'^a^  _  53^2  _  ^2/2 -f&y. 

28.  laW-lab.  43.    a^-3a^-a?  +  3. 

29.  8aj^-31a;2-4.  44.   i»2~2a;  +  l-/. 

30.  o?-l  +  2ab  +  h\  45.    a!*- 5  a;3  +  a;-5. 

31.  Qo^-Q.'c^.  46.    o?-h''  +  a-h. 

32.  a^/  +  .T2  — 7/  — 7.  47.    a^  —  172  a;  +  171. 

33.  (a  +  6)'-9(a  +  6)  +  20.        48.   3  a;^  -  3  a^  -  720  a^^ 

34.  3ax2-3a/  +  5  6a^-5&2/2.   49.    225 a;^ - 120 a^2/' +  16 2/^. 

35.  21x'^y  +  xy.  50.    a^ +  «'?>  + a&H  &^. 

36.  9»2_g2a;  +  9.  51.    x^ -{■{a-h)x-ah. 

37.  (a  +  &)'-8.  52.    a^  +  4«2-a;-4. 

38.  729a.'3o_100?/<^.  53.    x2  4-2a;  + 1 -4(a; +  1). 

39.  2mo^-Z4:^y  +  xy\  54.    a«-a. 

40.  128-2a2.  55.    aj« - 8 a^  + 16 a;. 

41.  10a^-33a3-7al  56.   36a2«-962«. 

57.  a^(a+&)+4a!(aH-&)+4(a  +  6). 

58.  a2-2a6  +  &'-6a4-66. 

59.  a2-4a&  +  362_6a  +  66. 

60.  a2-2a6  +  62_6a+66+5. 

61.  6a^-12a;y  +  6/-37a;  +  372/  +  6. 

62.  :^-f  +  o?-y\ 

63.  a2  +  62  +  c2+2a6  +  2ac  +  26c. 

Simplify  the  following  expressions,  and  factor  the  result : 

64.  (a;-l)(«-2)-6. 

65.  (a;+4)(aj-3)-18. 

66.  (a;  +  7)(a;-2)-(2a;-9)(3a;  +  2)4-(a?-3)(.c-10)-30. 


CHAPTER  VII 

HIGHEST  COMMON  FACTOR  AND  LOWEST  COMMON 
MULTIPLE 

HIGHEST   COMMON  FACTOR 

124.  The  highest  common  factor  (H.  C.  E.)  of  two  or  more 
expressions  is  the  algebraic  factor  of  highest  degree  common 
to  these  expressions  ;  thus  a®  is  the  H.  C.  F.  of  oJ  and  a^h''. 

Two  expressions  which  have  no  common  factor  except  unity 
are  prime  to  one  another^ 

125.  The  H.  C.  F.  of  two  or  more  monomials  whose  factors 
are  prime  can  be  found  by  inspection. 

The  H.  C.  F.  of  a^  and  \^b  is  a\ 

The  H.  C.  F.  of  a^foV,  a^hh'',  and  a'W  is  a^h^ 

The  H.  C.  F.  of  (a  +  hf  and  (a  +  hf{a  -  hf  is  (a  +  h)\ 

126.  If  the  expressions  have  numerical  coefficients,  find  by 
arithmetic  the  greatest  common  factor  of  the  coefficients,  and 
prefix  it  as  a  coefficient  to  H.C.F.  of  the  algebraic  expressions. 
Thus  the  H.  C.  F.  of  6  x'yz,  12  o?y%  and  60  x^  is  6  a^y. 

The  student  should  note  that  the  power  of  each  factor  in  the 
H.  G.  F.  is  the  lowest  power  in  which  that  factor  occurs  in  any 
of  the  given  expressions. 

EXERCISE  53 

Find  the  H.C.F.  of: 

1.  e>a%  2a^h\  5.  4  a  V,  ^a^^,  12  ax". 

2.  9a26V,  ISa^ftV.  6.  19  i»y,  95  ir^^  117  a;/. 

3.  17a^6%  51  cU  7.  12  x^y'z",  IS  xS/z^,  2^xYz\ 

4.  2^x?yz,  62/Va.  8.    Zy^fz,  9  a;V%  15a;y«. 

114 


HIGHEST  COMMON  FACTOR  115 

9.  4.a%h\  Ua'hH\  64  a^feV. 

10.  QSa^ft^c-^,  ISOa^^V,  -300a*6V. 

11.  15  a'bx'if,  -4:5  by,  -90  aH)^xY. 

12.  3(a  +  &)',  4  (a +  6)2,  3(a-f  &)Xa- &)• 

13.  3(aj  +  l)(a;  +  2),  12(a;  +  l)(a;  +  3),  6  (a; +  1)2. 

14.  (a  +  6)Xc  +  d)«,  (a  +  &)(c  +  d)^,  (a  +  b)\c  +  d)^ 

15.  6(a;  +  y)«,  ^{x  +  y)\x-y),  9(x  +  y){x-yf. 

16.  6  a2(a  +  b)\  8  a(a  +  6)«,  10  a^a  +  6)^ 


127.  To  find  the  H.C.F.  of  polynomials,  resolve  each  poly- 
nomial into  prime  factors,  and  apply  the  method  of  the 
preceding  article. 

Ex.  1.  Find  the  H.C.F.  of  x" - 4. xy -\- 4. y\  x'-Zxy  +  2y\ 
and  a^  —  7  ici/  +  10  y'^. 

a:2  -  4  a;y  +  4  2/2  =  (X  -  2  ?/)2. 

x^  -1  xy  -{■  IQ  y'^  =  (x  -2  y){x  -  b  y). 
Hence  the  H.  C.  F.  =x-2y. 

Ex.  2.  Find  the  H.C.F.  of  Qa^-^aW,  2  o?-^  o?b  +  &ab\ 
and  12  a^  - 12  a'b^ 

6 a*  -  6  ah^  =  Qa{a^-h^)=Q  a{a  -  h) (a'^  +  ab  +  62). 
2  a3  -  8  a26  +  6  a62  =  2  a(a2  -iab  +  Bb^)  =  2  a{a  -  6)  (a  -  3  6). 
12  a*  -  12  a262  =  12  a2(a2  _  52)  -  12  a2(a  +  6)  (a  -  6). 
Hence  the  H.  C.  F.        =  2  a(a  -  6). 

128.  If  7  a  is  contained  in  several  expressions,  obviously 
—  la  must  be  contained  also.  Similarly,  if  a  —  6  is  a  common 
factor  of  several  expressions,  —  a  +  6,  or  &  —  a,  is  a  common 
factor  also.  From  this  it  follows  that  each  set  of  expressions 
has  really  two  highest  common  factors,  whose  absolute  values 
are  equal,  but  whose  signs  differ.  E.g.  Ex.  1  of  §  127  has  the 
twoanswers,  a;  — 22/and22/— aj;  Ex.  2,2 a(a  — 6)  and  — 2a(a— 6) 
or  2  a(p  —  a). 


118  ELEMENTARY  ALGEBRA 

EXERCISE  64 

FindtheRCr.  of: 

1.  Six^y\  12a^f-18a^f, 

2.  12xyz',  SxYz'^-lSsc^y^z^. 

3.  36  a\  12aV  +  24a2ic3^ 

4.  9  ax- 12  a^ar,  24  a V  +  36  a^ic^^ 

5.  a'-b',  4a26  +  4a6l 

^       6.  4.d'-9b%  10  a^b-{- 15  a^ly". 

7.  dasc^-ieaf,  12  abx -^16  aby, 

8.  16a262-25  6V,  24a62-30  62c. 

9.  4.x^-]-12xy  +  9y%  16a;  +  24y. 

10.  m^  — 4a^,  m^  +  2mx. 

11.  m^  —  n^,  m^-i-miij  m^n-\-m'n?, 

12.  4a^  +  12ar^2/  +  9a;2/^  16 ici/ +  24  !/«. 

13.  9a3  +  24a26_^l6a62,  18a^  +  24.a^b, 

14.  4  a^ic^  _^  12  a^a;^/  +  9  aV,  18  a?x  +  27  a^ 

15.  a  +  6,  a^-6^  a^  +  fe^. 

16.  ax-\-ay  —  bx  —  by  and  a*  —  b\ 

17.  a^-7aj  +  12,  a;2-8a;  +  15. 

18.  a^-3aj-4,  a^-8a;4-16,  a^-16aj. 

19.  a2  +  3a;-18,  a3-27,  a'-6a  +  9. 

20.  a2  +  2a-3,  a24-7a  +  12,  a*  +  27a. 

21.  a^  +  3a;-54,  ar^  +  a;-42,  ic«  +  2aJ-48. 

22.  2a''  +  9a  +  4,  2a' -{-lla-^5,  2a'~-Sa^2. 

23.  Sa^  +  15a^b-\-lHab%  3a*  +  9 a^b  +  6a^b\ 

24.  a'  +  4.ab  +  3b\  a'  +  2ab-3b'',  a^ -{- 9  ab  +  IS  b^. 


HIGHEST  COMMON  FACTOR  117 

25.  j^-^xy-\-i:y\  x^-Sf,  x*  —  16y*. 

26.  a^-2x^-Sx-{-6,  2a^-5a;  +  2. 

27.  3a%-Sa'b-21ab,  Ta'-Ja-iQ. 

28.  a  — 6,  —  a  +  fe. 

'     29.    a  —  Sx,  3x  —  a. 

30.  13  a"  - 13  62,  26  b'  -  26  a\ 

31.  3a^-10a;  +  3,  9a;~-a^. 

129.  If  only  one  of  the  given  expressions  can  be  factored  by 

inspection,  determine  by  actual  division  if  its  factors  are  con- 
tained in  the  remaining  expressions. 

Ex.  1.   Find  the  H.  C.  F.  of  a^-4,  and  ic^-S  a^  +  8  x-12. 

The  factors  of  x^  —  4  are  x  —  2,  and  x  +  2.  By  dividing  x^  —  Zx"^ 
+  8  a;  —  12  we  find  that  a;  —  2  is  a  factor,  but  x  4-  2  is  not  a  factor.  Hence 
the  H.  C.  F.  =  X  -  2. 

130.  If  several  but  not  all  expressions  can  be  factored  by  in- 
spection, find  the  H.  C.  F.  of  those  which  can  be  factored,  and 
test  the  factors  of  the  H.  C.  F.  by  actual  division  as  in  the 
preceding  case. 

Ex.  2.   Find  the  H.C.F.  of  a^-3ar^-8a;H-24,  a?-21,  and 

a:8_3a;2_8a;  +  24  =  x2(x-.3)-8(x-3)  =  (x-3)(aj2-8). 
a<-27  =  (a;-8)(x2  +  3a;4-9). 
The  H.  C.  F.  of  these  two  expressions  is  x  —  3. 


x«-2x2  f  6X-27 

x-3 

x3-3x2 

X2  +  iC  +  0 

x2  +  6  X 

x2-3x 

9X-27 

9x-27 

Hence  the  H.  C.  F.  =  x  -  3. 

118  ELEMENTARY   ALGEBRA 

131.  If  divisor  and  dividend  are  both  arranged,  the  division 
can  be  exact  only  when  the  first  term  and  last  term  of  the 
divisor  are  respectively  divisible  by  the  first  and  last  terms  of 
the  dividend. 

Ex.  3.  Find  the  H.C.r.  of  a^  +  3a^-ll  a;-26  and  (3a;-l) 
(a;-3)(a;  +  2). 

Since  x^  is  not  exactly  divisible  by  3  «,  3  x  —  1  has  to  be  rejected,  and 
since  26  is  not  exactly  divisible  by  —  3,  x  —  3  has  to  be  rejected.  Hence 
only  X  +  2  has  to  be  tried.  But  actual  division  shows  that  x  +  2  is  a 
factor  of  x^  +  3  x2  -  11  X  -  26.     Therefore  x  +  2  is  the  H.  C.  F. 

EXERCISE  55 
Find  the  H.C.F.  of: 

1.  a'-3a  +  2,  a^-Ga'  +  Sa -3. 

2.  Aa^-l,  4:a^-6a^-4a  +  3. 

3.  a2-3a  +  2,  a'-\-a~2,  a'-^2a'-a-2, 

4.  2x'-}-5a^  +  2x',  a^-9a^-\-9x-70. 

5.  x^-4.x-\-3,  3a^-10x  +  S,  a^-Qx'-i-llx-e. 

6.  Sx^-1,  4.x'-l,  4.x^-\-x-l. 

7.  8x3-1,  8ic2  +  4a;  +  2,  16aj^  +  4x2  +  l. 

8.  a^-x,  x*-7  x^  +  6,  x^-3x^-\-5x^  +  Sx-6. 

9.  x^  +  5x-24:,  a^  +  4a^-26a;  +  15. 
10.    3  a^  +a  -  2,  4  a^  +  4  a^  _  a  -  1. 

(For  another  method  of  finding  the  H.C.F.  of  expressions 
which  cannot  be  factored  by  inspection,  see  Appendix  III.) 

LOWEST  COMMON   MULTIPLE 

132.  A  common  multiple  of  two  or  more  expressions  is  an 
expression  which  can  be  divided  by  each  of  them  without  a 
remainder. 

Common  multiples  of  Zx^  and  6y  are  SOx^j/,  60  xV?  300  x^y,  etc. 


LOWEST  COMMON  MULTIPLE  119 

133.  The  lowest  common  multiple  (L.C.M.)  of  two  or  more 
expressions  is  the  common  multiple  of  lowest  degree  j  thus, 
a^y^  is  the  L.  C.  M.  of  oc^y  and  xy^, 

134.  The  L.C.M.  of  two  or  more  monomials  whose  factors 
are  prime  can  be  found  by  inspection. 

The  L.  C.  M.  of  a^b^  and  a^b  is  a^¥. 

135.  If  the  expressions  have  a  numerical  coefficient,  find  by 
arithmetic  their  least  common  multiple  and  prefix  it  as  a  coef- 
ficient to  the  L.  C.  M.  of  the  algebraic  expressions. 

The  L.  C.  M.  of  3  a%^,  2  a^b^c^,  6  c^  is  6  a^b^ifi. 

The  L.  C.  M.  of  12  (a  +  by  and  (a  +  by  (a  -  by  is  12(a  +  by  {a  -  by. 

136.  Obviously  the  power  of  each  factor  in  the  L.C.M.  is 
equal  to  the  highest  power  in  which  it  occurs  in  any  of  the 
given  expressions. 

137.  To  find  the  L.C.M.  of  several  expressions  which  are 
not  completely  factored,  resolve  each  expression  into  prime 
factors  and  apply  the  method  for  monomials. 

Ex.  1.   Find  the  L.  C.  M.  of  4  a'b^  and  4  a^  -  4  a'b\ 
4  a262=  4^2252. 
4  a*  -4a263  =  4a2(a2_53). 
Hence,  L.  C.  M.  =4  a262(a2  -  &3) . 

Ex.  2.    Find  the  L.  C.  M.  of  a^  -  b^,  a'  +  2ab  +  b\  and  b-a. 

a'i-b^=:(a-\-b)(ia-b). 
a2  +  2  a&  +  &2  =  (a  4.  6)2. 

b  —  a  =  —  {a  —  b). 
Hence  the  L.  C.  M.  =  (a  +  6) 2  (a  -  6). 

Note.  The  L.  C.  M.  of  the  last  example  is  also  —  (a  +  by  (a  —  b).  In 
general,  each  set  of  expressions  has  two  lowest  common  multiples,  which 
have  the  same  absolute  value,  but  opposite  signs. 


120  ELEMENTARY  ALGEBRA 

EXERCISE  56 
Find  the  L.  CM.  of: 

1.  X,  x%  x\  .  5.   12,  15 a^,  25  ab^,  10  6^  ISa^fe. 

2.  X,  Sxy,  5xf.  6.    Sa^bx,  12  aV,  14/. 

3.  ic^  3a;2.v,  Saj;?^^  12.  7.    5a^/,  4/,  20  a^]/^,  45a;». 

4.  2a2aj,  Sft^y,  4a^ic2.  8.    13a;y,  15  a^?/^^  9a;y*,  c^ 

9.  4:a'bcd,  20  ab'cd^,  AOabc^d,  8d*. 

10.  2a^  4.a\  e>a'^,  h, 

11.  9  a;,  ^xy,  2{x  +  y). 

12.  4(a;-2/)^  ^(x  +  yf,  12(x--y)(x  +  y), 

13.  (a +  6)"^,  (a  +  5)'"+^  (a  +  6)'"+l 

14.  (a; -1)  (a; -2)2,  (a;  -  2)  (a;  -  3)^,  (x  -  3)  (a;  -  1)^. 

15.  3  a;,  5y,  x^y  —  xy^. 

16.  4a^,  32/',  6ary-6a^2/3. 

17.  6(a;  +  2/),  3x-3y,  6xy  +  6y'. 

18.  4a6c,  9a2-16ar^. 

19.  a'-b',  a^-b\ 

20.  a^  +  fe^  a?-ab-{-h\  a  +  &. 

21.  a;^4-5a;  +  6,  a^4-4a5  +  3,  a^  +  a?. 

22.  4.x'-4:x^,  3a^  +  6.T4-3,  6a^-12a;  +  6. 

23.  4a2  +  2a-2,  4a2-l,  4a2  +  4a-3. 

24.  3^2 _^3^^  5a^-5,  15a;. 

25.  a; +  1,  a; +  2,  a; +  3. 

26.  2a;-l,  4a;  +  2,  4a;2_i 

27.  2ar2-a;-l,  2ar2  +  a;-3,  4ar^  +  8a?  +  3. 

28.  1  — a;,  a;  — 1,  a;^  — 1,  a;^  — 1,  a^  — 1. 


LOWEST  COMMON  MULTIPLE  121 

29.  6x2-54,  7(x-Sy,  Sx'-9x-{-S. 

30.  x^  —  x^  +  x  —  lfa^  —  x. 

31.  3a2-lla  +  6,  3a2-8a  +  4,  a^-5a  +  6. 

32.  2£c3-3a;2-2a;,  2a^~15ar^- 8a;,  a^-10a;  +  16. 

33.  12a2+23a5  4-106^  4:d'-{-9ab -{-5b^,  3a'-i-5ab  +  2b^. 

34.  (a  — &)(6  — c),  (6  — c)(c  — a),  (c  — a)(6  — a). 

35.  (a  +  6)2-c2,  a2-(6  +  c)2. 

36.  x^-^1,  a^-2a^-{-4:X-S. 

(The  method  for  finding  the  L.  C.  M.  of  expressions  which 
cannot  be  factored  by  inspection  will  be  found  in  Appendix  III.) 


CHAPTER   VIII 

FRACTIONS 

EEDUCTION  OF   FRACTIONS 

138.  A  fraction  is  an  indicated  quotient ;  thus  -.  is  identical 

with  a-r-b.  The  dividend  a  is  called  the  numerator  and  the 
divisor  6  the  denominator.  The  numerator  and  the  denominator 
are  the  terms  of  the  fraction. 

139.  Siyice  a  fraction  represents  an  indicated  division,  the 
proofs  of  all  fundamental  properties  may  he  based  upon  the 
definition  of  division  ;  viz.  : 

a  r 

-  =  a?  means  a  =  bx. 
b 

140.  If  both  of  the  terms  of  a  fraction  be  multiplied  by  the 
same  expression,  the  value  of  the  fraction  is  not  thereby  altered. 

Expressed  in  symbols, 

0  mb 

According  to  the  preceding  paragraph,  this  really  means 

if  a  =  bx,  then  ma  =  mbx. 
But  in  this  form  the  correctness  of  the  conclusion  is  obvious. 
(§  90,  3). 

Hence  it  is  proved  that  ^=:^. 
b    mb 

141.  If  both  the  terms  of  a  fraction  be  divided  by  the  same 
expression,  the  value  of  the  fraction  is  not  thereby  altered ;  or 

ma  _  a 
mb     b 

This  follows  from  the  preceding  article. 

122 


FRACTIONS  123 

142.  A  fraction  is  in  its  lowest  tewns  when  its  numerator 
and  its  denominator  have  no  common  factors. 

6  ecu  z 
Ex.  1.    Reduce  ^   -^  ,  ,  to  its  lowest  terms. 

Remove  successively  all  common  divisors  of  numerator  and  denomina- 
tor, as  3,  X,  y^,  and  z^  (or  divide  the  terms  by  their  H.  C.  F.  Sxy'^z^). 

Hence  6xy^^2z^ 

143.  To  reduce  a  fraction  to  its  lowest  terms,  resolve  numerator 
and  denominator  into  their  factors,  and  cancel  all  factors  that  are 
common  to  both.  Never  cancel  terms  of  the  numerator  or  the 
denominator ;  cancel  factors  only. 

Ex.  2.    Reduce  ^'~^^'"^^,^  to  its  lowest  terms' 
6  a^  -  24  a" 

gg  -  6  gg  +  8  fl  ^  a(a2  _  6  g  +  8) 
6g4-24g2  6g2(g2-4) 

_    g(g  — 2)(g  — 4) 
~6g2(g  +  2)(g-2) 
g-4 
6g(g  +  2)' 

Ex.  3.   Reduce    ~:f  1+ ^^"?  if  2^T^x  *«i*«  lowest  terms. 

-12ai°(a2-121  W){a^-lr) 

-  6  g^  +  60  a%  +  66  g^ft^ -  6  a^jo?-  -  10  gft  -  11  h'^) 

-  12  gio(a2  _  121  52)  («2  _  52)      _  12  aio(g  +  1 1  &)  (g  -  1 1  &)  (g  +  &)  (a  -  6) 

^ -6g2Cg-ll&)(g  +  &) 

-  12  a}\a  +  11  6)  (g  -  1 1 6)  (g  +  6)  (g  -  6) 

1 

2  g8(g  + 11  &)(«-&)* 

144.  By  the  law  of  signs  in  division 

-\-a  _—a  _      -\-a_      —a^ 

That  is,  a  fraction  is  not  altered  if  the  signs  of  both  its 
t.erms  are  changed;  but  the  sign  of  the  fraction  is  changed  if 
the  sign  of  either  term  is  changed. 


124  ELEMENTARY  ALGEBRA 

Ex.  4.   Eeduce  (i^-^ab-2b^  ^^  .^^  ^^^^^^  ^^^^^^ 
W  —  o? 

a^  -i-ab-2  6'^ _a^  +  ab-2b^ 

62  _  o2  _  (^2  _  52-) 

a^  +  ab-2  b^ 
.  a^-b^ 

(a~b)(a  +  2b) 
(a-^b)ia-  b) 

-  _  «  +  2?>, 
a  +  b' 

EXERCISE  57 
Eeduce  to  lowest  terms : 

'   3af'  '    125 x^y  '    9x-9y 

2     48  a^b\  ^     17  a^+^  ^2  6  a6 


16  a^^^  16  a'*  6a26-6a62 

g     250  a^y^z^  g  35  a"+^6"'+^  ^^     aa;  +  ?>a?. 

25a^?/V'  '        Ta'b^     '                 '        ex     ' 

^     34  a^6V  ,   g  12  m  + 12  n  ^^     25ra;-35n/ 

85a^6V*  '      *  24:X  +  2Ay'              '    S5sx-4:9sy 

^      23  aV  ,^  9a2&+12a62                  4a2-9  62 

5.    -— - — -— r-r*  10. : -•  15. 


46  a'xY  9a^b-15ab^  2a  +  3b 

^g      4a^-96^  l  +  g^ 

(2a-36)2  •   Sa'-Sa^  +  Sa* 

17     fta^  +  ^g;  — ca;  gg     3  a;^  — 10  a;  +  3 
'   ay  -{-by  —  cy  '     9a^  —  6a;  +  l 

.^     m^-2mn-{-n^  -.     4:  a^ -\- 12  ab -^ 9  b^ 
m^  -  ri"  16  tt6  +  24  b^ 

19     9a^  +  6a;  +  l  04     ci^  -  7  a  + 12 
9a}^-l  '   a2-8a4- 15* 

^^^     aW  +  1  .  2g^    a«-3a-4 


a^b'c'-^abc  d^-Sa-^-ie 


FRACTIONS  125 

^  •    a2-2a-3  '    2i>3-5i)2g  +  3i)<?2 

2a2-f-9a-h4  a^4-?>'  +  2a6-c^ 

2a2  +  lla  +  5  *    a^ -f- c^  +  2  ac  -  6«* 

15a^  +  2a;y-/  3.     3  a;^  -  21  a^  +  36  a:^ 

•   Sm'-7mn-i-2n^'  '   a^  +  b^' 

^^     ax  +  ay  —  cx-cy  ^^     a^  —  h*^ 

'   ax -\- ay  -{■  ex -\- cy  a^  —  b^ 

31     2ac-2ad-Sbc  +  Sbd.  3^     (a  +  &y. 

2ac-2ad-\-Sbc-3bd  '    a^-i-b^ 

32^    ma;  +  m-a;-l  ^^       lp(f-^lpq 

m^  —  1  *    14^9^'*  — 14jp5 

mn  —  2m4-3yi  —  6  .-     a  — 6 

m?i  —  2m  —  39i  +  6  6  —  a 

a^_4aj  +  4           ^,     a^-Oa^  +  Uft^                 a«--l 
! .         43.    1 .         44.    . 

_a^  +  5a;-6  262  +  a6-a2  1-a^^ 


42. 


145.  If  only  one  of  the  terms  of  the  fraction  can  be  factored  by 
inspection,  find,  if  possible,  the  H.  C.  F.  according  to  §  129, 
and  divide  numerator  and  denominator  by  it. 

Ex.  1.   Reduce  ^^'~^^^'  +  ^^  ^~^^  to  its  lowest  terms. 
9  a^  -  36  a  +  35 

9  a2  -  36  a  +  35  =  (3  a  -  5)(3  a  -  7). 

But  3  a  —  5  cannot  be  a  factor  of  the  numerator.     (Art.  131.) 
Hence  only  3  a  —  7  need  to  be  tried. 

Actual  division  shows  that  3  a^  _  13  ^2  _|.  23  a  —  21  is  exactly  divisible 
by  3  a  —  7,  and  that  the  quotient  is  a^  —  2  a  +  3. 

Therefore  3  a«  -  13  a^  +  23  a  -  21  ^  (3a  -  7)(a2  -  2  a  +  3) 
9  a2- 36  a +35  (3  a  -  5)(3  a  -  7) 

_  g^  -  2  g  +  3 
3o-6 


126  ELEMENTARY  ALGEBRA 

146.  Since  a  factor  of  each  term  of  a  fraction  is  also  a  factor 
of  their  difference  or  sum,  we  can  frequently  find  the  H.  C.  F. 
of  these  terms  without  factoring  them,  and  thereby  reduce  the 
fraction. 

Ex.  2.   Eeduce  2  a^-4a^  +  5  a;-33  ^^  .^^  ^^^^^^  ^ 

The  difference  between  numerator  and  denominator  =  3  x^  —  4  a;  —  15 

=  (3x  +  5)(x-3). 
3  a;  +  5  is  not  contained  in  either  term.     (Art.  131.) 
Hence  if  there  is  a  H.  C.  F.,  it  is  x  —  3. 
By  trial  a;  —  3  is  found  to  be  the  H.  C.  F. ,  and  the  fraction  reduces  by 

division  to  2^^+1^+il. 
2  a;2  -  x  +  6 


EXERCISE  58 

Eeduce  to  lowest  terms : 

a^-2x'  +  7x-S0  ^     a^  +  5a^b-\-Sab^-\-6 

D. 


Saj^-lO.T  +  S 

x'-Tx  +  e 

'2a^-5i^  +  2x 

a^_5a:2_8^_^48 

a^_9a;''^_{_20a; 

a;3_|_6a^_l_7a;_(_10 

ar^  +  5a^_a;_5 

a^_6a^  +  lla;-6 

3.         „  ..     Z7      '  8. 


9. 


a^-^3a^b+4.ab^-{-2b^ 

Sa^  +  7a'-Sa-2 

Sa'^2a'-10a  +  5 

x^-4.x^-{-x  +  e 

a^-3x'-{-Sx-2 

5o^-Sa^y-^2xif-4.f 

5i^-9x'y-{-7xy'-3f 

2'x'  +  x'-2x-l 

5. '  -'  10. 


1 47.  Reduction  of  fractions  to  equal  fractions  of  lowest  common 
denominator.  Since  the  terms  of  a  fraction  may  be  multiplied 
by  any  quantity  without  altering  the  value  of  the  fraction,  we 
may  use  the  same  process  as  in  arithmetic  for  reducing  frac- 
tions to  the  lowest  common  denominator. 


FRACTIONS  127 

Ex.  1.   Eeduce  — ^,  — -,  and  ■— ^  to  their  lowest  cora- 
j  •     ^       b  6  V    3  a^  4  ab^    * 

men  denominator. 

The  L.  C.  M.  of  6  6%^  3  a^,  and  4  afe^  is  12  a^ftSaja. 

To  reduce  — ^  to  the  denominator  12  aWx^^  numerator  and  denomi- 

^  ^'^^                 12  a^'^x^ 
nator  must  be  multiplied  by  or  2  a^ 

Similarly,  multiplying  the  terms  of  — -  by  4  h'^x'^,  and  the  terms  of 
1^  by  3  a^^2,  we  have  -ii«^,    -«^,  and  i^^^. 

148.  7b  reduce  fractions  to  their  lowest  common  denominator^ 
take  the  L.  C.  M.  of  the  denominators  for  the  common  denominator. 
Divide  the  L.  C.  M.  by  the  denominator  of  each  fraction^  and 
multiply  each  quotient  by  the  corresponding  numerator. 

Ex.     Reduce . ; -->  and -— — 

(a;-f3)(iB-3y  (a;-3)(a;-l)  (x-l){x+3) 

to  their  lowest  common  denominator. 

The  L.C.D.=(a;  +  3)(x  -  3) (a;  -  1). 

Dividing  this  by  each  denominator,  we  have  the  quotients  (aj  —  1), 
(x  +  3),  and  (x  -  3). 

Multiplying  these  quotients  by  the  corresponding  numerators  and 
writing  the  results  over  the  common  denominator,  we  have 

3x2 -3x  3x^  +  7a;-6  ^^^  5a; -15 

(x+3)(x-3)(x-l)'    (x+3)(x-3)(x-l)'  (x+3)(x-3)(x-l)* 

Note.    Since  a  =  -,  we  may  extend  this  method  to  integral  expressions. 
EXERCISE  59 

Reduce  the  following  to  their  lowest  common  denominator : 

2a2    5^  36_^  5     a^     1       4 

'   Sx"'  4a2*  •     '   aY  '   a^   o?'   3a*' 

12  x"       7  4    1    1    i  6    J^     l^^ 

*   b<^b'''   Sab''  '   a    b'   0  '  4:(^d'   2c'd^' 


128  ELEMENTARY  ALGEBRA 


7. 

6 

10          ^                ^ 

a;^  —  ax    a?  —  o?x 

p 

3cv« 

5ar*/      2.'C« 

3a  +  2    20.24-1 

Ta^dV 

2a^6V  3a6c 

'   3a-9'     a2-9 

n 

1 
a-6'    . 

a 

1„        3           5            6 

a^-a6 

a;4-a    x-a     o^-a? 

13. 

3a 

4a                6a 

o?-{-2ah  +  h'^    a^- 

-2a6  +  62'    a'-h^ 

14. 

5x^             1 

5 

^^^    a5^_2a;-3' 

cc2-9 

15. 

a;  4-  3               a; 

+  2              oj  +  l 

a;2_3aj4_2'    a;2_ 

4a;-f-3'    a;2-5a;  +  6 

1A 

p               6p 

9p 

p-q    p'-hpq  +  q 

,'     ^3_^ 

17. 

Ap               9 

_  ^q  . 

p-hq    I^-pq  +  q"'  P^  +  (f 

18. 

a2  +  62        26 

ab 

2ab  '   a^-ab'    ab-b^ 


ADDITION  AND  SUBTRACTION  OF  FRACTIONS 

149.   Since  -  4-  -  = (Art.  74),  fractions  having  a  common 

c      c         c 

denominator  are  added  or  subtracted  by  dividing  the  sum  or  the 
difference  of  the  numerators  by  the  common  denominator. 

160.  If  the  given  fractions  have  different  denominators, 
they  must  be  reduced  to  equal  fractions  which  have  the 
lowest  common  denominator  before  they  can  be  added  (or 
subtracted). 


FB  ACTION  8  129 

Ex.  1.    Simplify       2a  +  36  Qa-b     ^ 

The  L.  C.  D.  is  4(2  a  -  3  6)(2  a  +  3b). 

Multiplying  the  terms  of  the  first  fraction  by  2(2  a  +  3  6),  the  terms  of 
the  second  by  (2  a  —  3  6)  and  adding,  we  obtain 

2a  +  3&  6a -b     ^2(2  a  +  3  6)(2  q  +  3  6)  +  (2  a  -  3  6)(6  «  -  &) 

2(2  a -36)      4(2  a +  36)  4(2  a  -  3  6)  (2  a  +  3  6) 

^  8  a2  +  24  a6  +  18  62  +  V2  gg  -  20  a6  +  3  b^ 
4(2a-36)(2a  +  36) 
20  ff2  +  4  a5  +  21  62 
4(2a-3  6)(2a  +  3  6)' 

151.    The   results   of  addition   and   subtraction   should   be 
reduced  to  their  lowest  terras. 

-n>     «     o-      ^•£     a4-2b  ,         2a  +  6  a  —  Sb 

a^  —  ab  —  a{a  —  6), 
a2_3a6  +  2  6-2=(a-6)(a-2  6). 
oP-  -  2  ab  =  a{a  -  2  6). 
Hence  the  L.  C.  D.  =  a{a  -  6)  (a  -  2  6). 

q  +  26  .         2a  +  6 a  -36 

a2_a6     a2-3a6  +  262     a2  _  2  a6 


_  (a  +  2  6)  (a  -  2  6)  +  a(2  «  +  6)  -  Ca  -  3  6)  (a  - 

-6) 

a(a-6)(a-2  6) 

_  ^2  _  4  62  +  (2  a2  +  ab) -(a^-iab  +  3  62) 

a(a-  6)(a-2  6) 

_  a2  _  4  ft2  +  2  a2  +  a6  -  a2  +  4  a6  -  3  62 

a(a-6)(a-2  6) 

_  2  a2  +  6  rt6  -  7  62  _  (2  a  +  7  6)Ca  -  6) 

a(a  -  6)(a  -  2  6)      a(a  -  6)(a  -  2  6) 

^  2a  +  76  _ 

a(a-2  6)' 

Note.  In  simplifying  a  term  preceded  by  the  minus  sign,  e.g. 
-(a  —  3  6)(a  —  6),  the  student  should  remember  that  parentheses  are 
understood  about  terms  (§  68) ;  hence  he  should,  in  the  begyining,  write 
the  product  in  a  parenthesis,  as  —  (a2  —  4  a6  -f  3  62). 

K 


130  ELEMENTARY  ALGEBBA 


EXERCISE   60 

Simplify : 

5a-7b     3a-5b     lla-35 
4  6  12       ' 

3a  +  46     5a-7b     6a-\-llh 


2. 


15  45 


-     a  ,  c  -o  a-\-h     a  —  b  ,  a  — 2b 

o» r"~*  i.o. 1-  ■                « 

b     d  6a         56  2a 

4.    l-l.  14.  a;-2?/      a;-?/  ^  a;+y. 

'    X     y  '      2Qi?         2xy        Z'if' 

^     5 be  ,  Sac  Tab                   -^        1      ,      1 


3a       46       2c  a;-|-3a;-l-4 

^     2aH-36      3a-7b  ,«        1  1 

4a  56  x  —  3     cc  — 4 

6a;-ll?/     15y  — 8  a;  ^^         5  6 

13a;       .        172/      '  *    a;-l^  +  l* 

8.    7a;4-5y     2y-3g  ^    4y  ^^         x  y 

5a;^  1  yz         Sxz  x-\-y     x  —  y 

^      2    .  9a-26      a+6  ^^        Aa  36 


ha        15a6         lOa^  a-26      a-36 

lu—3v    5u-\-6v    5uv—Sv  _       1  +  a;     1  —  a; 
12 wv        18 V          30w    *        *    l-x     1+a;' 

11.    ^4.-.     (HiKT:Leta;=^.^  21.    ^^  +  ^^^il/. 


M 


1/  a;  +  3?/     x  —  Sy 


12     2a  +  3       a       3a-26  x  +  y      a;-3?/     ar^-4y^ 

•       6tt2    "^262"^      6a6  *    x-y       x-{-y  '^  x'-y' 

2a-36     2a  +  36      8a^  +  186^ 
'    2a  +  36     2a-36      4a2-962' 

.       24.    ^^  +  .V'     4a;-y 
'    4:X^  —  y^     4:X  +  y 


FRACTIONS  131 


2-       5x         6x         x^  —  2x-{-l 


x-9     x-3    ■^-12x-\-27 

26.  ^'     +       ^       +2. 

27.  «        I     «  +  ^  & 


a2  +  a6      a6-52     a^-h^ 
28.    i ^+   «  +  2 


a     2a— 4     3a— 6 

29.     ^— ^,  +  ^. 

30     _1 ^  «^  +  2 

a-1      a^  +  a  +  l      a'-l 

2 


31.    a— 1  + 


a-f-1 


32.    a +  6         ^' 


a  — 6 

33       a;-4       3a;-5  5a;^  +  9g!  +  14_ 

2a;-.l       a;  +  2  2ar^4-3a;-2' 

34.  -±--^ 1-.  36.     _^  +  ^-^. 

a;  —  1      a;  —  2     a;  —  3  x-\-m     x-\-h     x  +  c 

35.  ^— -^__^.  37.    ^+^+    2 


a;  — a     x  — 6     x  —  c  {x—Vf    (a;— 1)^    a;— 1 

16  1  19 


38. 


39. 


9(a;-7)     4(a;-2)      36(a;4-2) 

a . h 2b  — a 

a? -2  ah  +  h^     o? -\-2  ab ^W      a^-b^' 


40„    ^  +  ^^y     <a?-.v)        2a:y 
^y  —  y^      xy  +  x^      x^  —  y^ 

41.     ^-1 +  -^-i +  1 


ar'-3a!  +  2     x'^ix  +  S     !>^-5x  +  6 


132  ELEMENTARY  ALGEBRA 

42     cl'  +  <^^  +  ^'     a'-ah  +  W 


43. 


a+b  a—b 

3  m  3m 


1  —  m^     1  4-  m^ 


44.  — ^ h 

a;3  +  a.y     x^y  +  f     xy{x-\-y) 

^^2  3,3  a; 

45. 1 


l-3a5     l-2a;     a;     l-Sa^  +  GiC^ 

46.    -.^--^      (a+b), 
of  —  ab     a^  +  ab 

4.  o?          ,         25  62           9a     5b 
47. , 

10  a6  -  25  6^  ^  4  a^  -  10  a6     5  6     2  a 


48. 


49. 


2a;-l 2a;  +  l 9(cc^4-2) 

4a^4-4a;  +  l     4a;2_4^^i      16  a;^  -  8  fl;^  4.  l 

_J. 1  4  Sa^ 

a_2     a  +  2      a2_^4     a*-l' 


Solution 
Combining  the  first  two  fractions, 

1  1  4 


a-2a  +  2a2.-4 
Combining  (1)  with  the  third  fraction, 


(1) 


'  +-A-.-4^.-  (2) 


a'^-4      a2_^4     a*  - 16 
Combining  (2)  with  the  last  fraction, 

8a2  8a2  120^2 


^4  _  16     a4  _  1      («4  _  16)  (a*  -  1) 
50.   -1—-?-+     4^ 


(3) 


a-|-6     a  —  6      o?—b^ 

-,     1+a;     1  — oj.l  +ar^ 

ol. 1 • 

1  — a;     1  +  a;     1  — a^ 

52    3|)  +  4g     3/>-4^_      24m     , 
•   3j9_4g     3i)  +  4<y      9y_i6^ 


FRACTIONS  133 


53.   ^  +  -i^+    2« 


a  +  l     a-1     a^  +  l      a^-1 
54.    — ■ h 


x  +  y      x  —  y      x^  —  y^ 

^^        1      ,      4&  1  26 

oo. + 


a  +  6     a^-6=     a-6     a'^  +  ft'^ 
56.^-^^ ,  +  r, ^TT— .+ 


(a  —  b){a  —  c)      (h  —  a)(b  —  c)      (c  —  a)  (c  —  6) 

Solution.     In  examples  of  this  type  it  is  advisable  to  arrange  all  factors 
in  the  denominator  in  a  similar  manner  ;  here  in  alphabetical  order. 

Instead  of  (&  —  a),  write  —(a  —  b). 
Instead  of  (c  —  a),  write  —(a  —  c). 
Instead  of  (c  —  &),  write  —  (&  —  c). 

Substituting  the  values  and  considering  §  144,  the  expression  becomes 
a  b  ,  c 


(a  -  b)(a  —  c)      (a  —  6)  (6  —  c)      (a  —  c)  (&  —  c) 

The  L. CD.  is  (a  -  6)(6  -  c)(a  -  c). 
Hence  the  expression  equals 

a(b  —  c)—  b(a  —  c)  +  c(a  —  h)  _Q 
(a  —  6)(&  —  c)(a  —  c) 

60.   ^ 2_^       15 


57. 

2     +     3    . 
1-a;     x-^l 

58. 

6a          6b 
a—b      b—a 

59. 

.^.+  ^  . 

a; 

61. 


2a-8     3-2a     9-4.a^ 
S-{-x~3-x~  a^-9 


62.   a-f--^--^-f     ■'■ 


ar^  — 1     a;  +  l     1  — a;  a  +  1     1  — a     a  — 1 

63.   a2  +  a  +  l  +  3-^^ —• 


a     a 


64.  ?!±l-(.-l)-^  +  ^ 

a;— 1  1  — a;a;2  — 

65. J- -  +  — 


(a  —  6)(a  —  c)      (6  —  c)(6  —  a)      (c  —  a)(c  —  6) 


134  ELEMENTARY  ALGEBRA 


66.    , ^^ ,  +  7^ h-. .+ 


67. 


(a  — 6)(a  — c)      (b  —  a){b  —  c)      (c  — a)(c  — 6) 
x  +  S x-2 


68.         ^-^       +      "^"^^  "^-^ 


69.  2a;4-l 


2a;-l 


70.  2.  +  ^^^:zM±l_i. 

4aj2-4x  +  l 

152.  Reducing  mixed  expressions  to  improper  fractions.  Since 
an  integral  expression  may  be  written  in  the  fractional  form 
with  the  denominator  1,  this  is  merely  a  special  case  of  addi- 
tion  of  fractions.      E.g.  a4--  =  -  +  -=  — ^^.      Hence    the 

c     1      c  c 

numerator  of  the  required  fraction  is  obtained  by  multiplying 
the  integral  expression  by  the  denominator  of  the  fraction  and 
adding  (or  subtracting)  the  numerator  of  the  given  fraction. 

EXERCISE  61 
Eeduce  the  following  expressions  to  fractional  form  : 
^  7.   ^-3  +  ^"^-^ 


1. 

a 

c 

2. 

^+1+1. 

x—1 

3. 

4. 

a     3     «'  +  « 
a-2 

5. 

5        3ab-7b 

^"           26      • 

6. 

a«-l-«'-l. 

8.  a^  —  a  +  l- 

9.  a-1  — 


a-2 
a  +  1 


a-1 
a  +  1 


-a  +  l 
10.   a^  +  a  +  1- 


2  I  ^  I  -I      a^  —  a-^1 


11.   3x-y  — 


a 

10  a;?/ 
x-Sy 


a^-2b'  ,  ^      o;, 

12. — ta  —  oo, 

2a-Sb 


FB  ACTIONS  135 

153.   To  reduce  a  fraction  to  an  integral  or  mixed  expression. 

^  5a^-15a-7     Bo"     15a       7  „       7 

Hence = — -  =  a  —  3  —  — -. 

oa  oa       oa      oa  oa 

Ex.  1.     Reduce — ^^-— to  a  mixed  expression. 

2a;  — 3 


2x-3 


2a;2  +  2*ic  +  6 


Therefore 


4a;3-2ic2  +  4a;-17 
4a;3-6a;2 

4  x2  +  4  a; 
4a;--^- 6a; 
+  10a; -17 
+  10a; -15 


4a;3-2x2  +  4a;-17^(2x24-2a;  +  5)(2a;-3)~2 
2a;-3  (2a;-3) 

=  2x2  +  2a;  +  5  ^ 


2a;-3 

A  fraction  may  be  reduced  by  division  to  an  integral  or 
mixed  expression,  if  the  degree  of  the  numerator  is  equal  to,  or 
greater  than,  that  of  the  denominator. 

If  the  remainder  be  a  polynomial  whose  first  term  is  nega- 
tive, it  is  customary  to  write  a  minus  sign  before  the  fractional 
part  of  the  result  and  to  change  all  the  signs  of  its  numerator. 

^  2a^-^4:0(^-3x  +  S      o  .     -2x^-^5x-S 


ii^  +  3x'-4:X-\-3  a^-^Sx'-4:X-h3 

^o         2x'-5x-\-3 
a^4.3iB2_4a;  +  3 

EXERCISE  62 

Reduce  each  of  the  following  fractions  to  a  mixed  or  integral 
expression : 

6ar^-9a;  +  8  6a;^  4-8a;^- 4a; -7 

3a;        "  *  2ar^  * 


136  ELEMENT ABY  ALGEBRA 


x—'d 

x  +  2 


a«  +  3a25  +  3a6^4-2 

a  +  6 
2a^_6x  +  17 

2x-4. 

a^^-y  +  2 
a  +  b 

x-\-l 

6.  2l±i*!.  10. 

a-26  a^^a  +  1 


MULTIPLICATION^  OF  FRACTIONS 

154.  Fractions  are  multiplied  by  taking  the  product  of  the 
numerators  for  the  numerator,  and  the  product  of  the  denominor 
tors  for  the  denominator  ;  or,  expressed  in  symbols: 

If  x  =  -  and  ?/  =  -, 

b  ^      6l 

then  xv  —  — 

^     bd' 

According  to  §  139,  this  really  means : 

If  bx=^a  and  dy  =  c, 

then  bdxy  =  ac. 

But  in  this  form  the  correctness  of  the  conclusion  is  obvious. 

(§  90,  No.  3.) 

Hence  it  is  proved  that 

ac  .  a       -i         c 

xy  =  -— ,    or  since   x  =  -  and  y  =  -, 

^     bd'  b  ^     di 

a     c  _aG 

b  '  d~bd 

155.  Since  -  =  a,  we  may  extend  any  principle  proved  for 

fractions  to  integral  numbers,  e.g.  -  x  c  =  — 

b  b 

To  multiply  a  fraction  by  any  integer,  multiply  the  numerator 
by  that  integer. 


FRACTIONS  187 

156.  Common  factors  in  the  numerators  and  the  denominators 
should  be  canceled  before  performing  the  multiplication.  (In 
order  to  cancel  common  factors,  each  numerator  and  denomi- 
nator has  to  be  factored.) 

Ex.1.    Simplify        __x  — X-^. 

The  expression  =  -  4  •  6  .  7  gB^cWy 
3  .  5  .  2  .  a  •  ab-^cd^x 

__  28 acx^y 

Ex  2    SimBlifv  <^^— ^^         ,        a^^4tah'^  6a— 186 

^     ^    a2-5a6+662  *  3a2+3a6-h362  '  7(a-6)2' 

^  (g  -  ft)(q2  +  ah  +  h^)      a(a  +  2?))(a  -  2  6)  ^  6(a-3  6) 
(a -2  6) (a -3  6)       *      3(a2  +  a6  +  62-)      *   7(a-6)2 

_2a(a  +  2&)^ 
7(a-6)    * 

EXERCISE  63 


j'ind  the  following  products 

mV      24 
*     36      mhi^ 

4. 

x'f      lah 
7az^     ix^z 

„     a6V      3d3 
*    2d^     5a'b^ 

5. 

Sa^f    6yh^    10  aV 
Sz*       5a^        92/2 

*     15  z'      21  a^/ 

6. 

12m  +  12n      r^  +  r 
24  ?*  +  24  r     m  4-  n 

^     4a2-962 

1 

2a  +  36   •  2a-36 
.     4a2_f.l2a64-962  4a3 


3a^  (2a  +  3by 

169     a^-Sa  ^^  5  a  a^-Qa-hO 


a^-64:       a-13  a^-5a  +  6      5a:'-^5a 


138  ELEMENTARY  ALGEBRA 

11.    «  +-^  .  <^  +  2  ^g     a;2-5a;  +  4  ^  a;^  +  3a;--4 

-„     x2-l    a^-4a?+3         7  ,^     a^^  +  a^^    a^-a^a 

1/0.     • • •    14.     ■ •  • 

a;+3        x'+x        (x-lf  o?h-aW    a^-^ab^ 


15. 


16. 


x'-ex-^B     5x^-5 


(x-iy        6x-{-6  X 

a2  +  18<x4-80       a2  +  6a-7 


a^-\-ba-bO     a'  +  lba  +  m     a 


j^     a^-8a    4&^-86+4  5  +  2 

262^2  *       o?-2a       '  a24-2a  +  4* 

a  —  6\2   a^  4-  a& 


19.    ^^^Y.^!zl4.  21.    f_i_Y.(^±i> 


a-\-bJ    ab  —  W 

3    a:2__49 
16 


DIVISION   OF   FRACTIONS 


157.   Divide  -  by  -• 
b^  d 


Let                                                       a;  =  ^- 

b 

d 

Then,  by  definition  of  division,  x  '-  =  -- 

d     b 

Multiplying  both  members  by  -,      x  =  -  > 

c              b 

A, 

c 

Or                              ?^-,=r 

b      d      b 

A. 
c 

158.  To  divide  an  expression  by  a  fraction,  invert  the 
divisor  and  multiply  it  by  the  dividend.  Integral  or  mixed 
divisors  should  be  expressed  in  fractional  form  before 
dividing. 


FRACTIONS  139 

159.   The  reciprocal  of  a  number  is  the  quotient  obtained  by 
dividing  1  by  that  number. 

The  reciprocal  of  a  is  — 
a 

The  reciprocal  of  |  is  1  -j-  |  =  |. 
The  reciprocal  of is  1  h 


a  +  h 

Hence  the  reciprocal  of  a  fraction  is  obtained  by  inverting 
the  fraction,  and  the  principle  of  division  may  be  expressed  as 
follows : 

160.  To  divide  an  expression  by  a  fraction,  multiply  the 
expression  by  the  reciprocal  of  the  fraction. 

Ex.  1.   Divide    ^^-f    by    ^-y\ 
ar  4-  xy^         xry-\-y^ 

7?  —  y^  ^2   _   y1     ^    a;3  _   y3  y^y  _[_  yS 

ac*  +  x]j^  '  x'^y  +  2/3     x^  +  xy'^  '    x^  —  y^ 

^(x-y)  (x^  +  xy  +  y2)  ^      y (a;^  +  y"^) 

x(x'^-^y^)  (x  +  y)(x-y) 

_y(x^  +  xy  +  y^)  ^ 
""      xix  +  y) 

EXERCISE  64 
Simplify  the  following  expressions : 

1  i«!5^ia^  5     a^  +  ab  ,  a^ -{- a^b 

'   Sx'y  '  3xy'  '   a"  -  ab  '  ab^ -{- b^' 

2  17  a'b*  .   3  a'b^  ^    a^-Sa-\-2  .  (a-2y 

7xy^    '  Ux^/  '   a2-5a  +  4  '  (a-l)2* 

3.   ^-^^.  7.   l^^-Ili^^2a-36. 

/  2  a  -f  3  6 

4     9 a  .  ^  a4-5&    .  ab  +  5b^ 

*     6*  ""   *  '   a2  +  6a6  *  a^-\-6a'b' 


140  ELEMENTARY  ALGEBRA 


10. 


o}^h^   ^  (a- by 
(a  +  by'      2  a 


11     «^-5a     a'  +  a-20  .  a2-2a-8 
a-1  a'-25      '    a^  +  a-2  ' 


12. 


a^-b^    ^  a2^ab^b\ 
a^  +  a'b  '     aV'  +  d'b 


13  2ct^-13a  +  15     2a  +  l  .    a-5 

4a2-9  2a-l  '  2a-l' 

14  3?)^     ^^^a(CT  +  2  6)  .     2a& 
a2-a6        ab  +  W     '  a^-b'^' 

15.      ^-^    ^  6  a^ft  -  4  a62  ^26^ 


a  +  6     45a2-2062  '  15a  +  105 

16  6a^-a-2   ^,  2  a^-5  a-12  .  2  6X^-7  a-4 
4a2  +  4a-3       9a2  +  6a-8    '6a2  +  5a-4' 

17  a'-2a  +  4^^^^  a2  4-a-2    ^        a^  +  8a 

a_5  a2-2a  +  l  '  a3  +  4a2-5a* 


18  («  +  ^)'.     ^'-^' 
a-36   *  a^-21b^' 

19  .a?  +  y  y  ^  +  <y  +  a^y^  .  a^  —  3  a;y  +  2  y'^ 
'    ^-f  {x  +  yf       '          {x  +  yy 

20.    ^-^^y  y^  +  ^y  +  y^  .  a^  +  a^V  +  a?/ 
x^  —  xy  x-\-y        '         x  —  y 

21  49a?»y^-^-^      (a--2y\  (a-2)» 

a2_4  7aj2/  +  ^'  *  («  +  2/ 

22  o^'-Q>-cy  ,  {b-cy-a? 
c^-ip-ay  '  (a-by-c" 


FRACTIONS  141 


COMPLEX  FRACTIONS 

161.   A  complex  fraction  is   a  fraction  whose  numerator  or 
denominator,  or  both,  are  fractional. 

Ex.  1.    Simplify  - — ?— ^. 
h      c     a 

6      c     a  ahc 


h      c     a  ahc 


ah -\- ah^ -\- hc^  ahc 

X 


ahc  ah  —  ah^  +  ha? 

^ah  +  ah^  +  hc^ 
ah  —  ah^  +  hc^ 

162.  In  many  examples  the  easiest  mode  of  simplification  is 
to  multiply  both  the  numerator  and  the  denominator  of  the 
complex  fraction  by  the  L.  C.  M.  of  their  denominators. 

If  the  numerator  and  denominator  of  the  preceding  examples 
are  multiplied  by  ahCj  the  answer  is  directly  obtained. 

x-\-y  ^  x-y 

Ex.  2.    Simplify  "^T"^     "^t^- 


x—y     x+y 

Multiplying  the  terms  of  the  complex  fraction  by 
(x  +  y)(x — y),  the  expression  becomes 

(x-{.yy-{-(x-yy^2x^-{.2f 
(x  +  y)-(x-y)  2y 

^^  +  y\ 


142  ELEMENTARY  ALGEBRA 


Simplify : 

EXERCISE  65 

a 

5. 

a 

c 

-5 

c 

6. 

c 

c 

a 

3.    I 
c 

d 

7. 

a      c 
b     d 
a      c 
b'^d 

a  +  ^ 

1           r 

8. 

1       l' 

^-       d 

c     a 

2a'     o. 

13.    -^ — .  17- 


2a 
26 

-1 

1 

1 

1 

a  +  1 

a 

-1 

1 

1 

a-1 

a 

+  1 

X 

y 

y 

X 

1^-    -^ --•  18. 


15. 

y    X 

V       X 
16.    -^ 20. 

^  +  1-2 

V        X 


a  +  b 


9. 


10. 


11. 


1 

,  1 

+  - 

c 

a 

,  1 

a 

+  - 

a 

1 

a 

a 

a 
5 

+  4a; 

a 
5 

+  5a; 

12. 

3a- 

1  + 

3a^ 
3a 

2x 
a 

a; 

+  1        X 

-1 

iC 

-1      aj 

+  1 

X 

+  1  ,x 

-1 

X 

-1  'a; 

+  1 

2 

a  — 5  6 

, 2a+56 

2 

a-f  56 

'  2a- 

-5b 

2 

a  +  56 

2a- 

-5b 

2a- 

-5b    : 

s+ 

^  +  1 

a; 

b' 

x' 

^+1 

X 

Sx'-^x-2 

1 

2 
Sx 

2a-{-5b 


FRACTIONS  143 


^^y 

26. 

a 

oi     y   ^ 

\yJ 

i+- 

a 

1         1 

x-2     x-S 
22. 

27. 

5+1 

1   1             ^ 

^4 

'  a^_5a;  +  6 

23.    ^          ^;^      1+^^ 
a; 

28. 

a 

•+i 

x_^y     x_y 

29. 

1 

24.    -^      %^     ^. 
2/     a;     2/     a; 

a 

a;  1      ^         a;     2-|-^ 

30. 

1 

ic  — 3                   a; 

OK                                                V                                        ■ 

„,         a  +  1 

X ^     aj-3  +  i 

a;— 2                  X 

REVIEW  EXERCISE  II 

Simplify : 

'■  («-)(M)- 

4. 

\l2y     16 As 

12a!«\ 
25  W 

'■  (lf-=")Kii-"> 

5. 

(-')ei)- 

3.  {i-KM,uy 

6. 

(f-')e-> 

■  '-e-De-^ 

8.    ^2a_36\     /_ 
1,36     2a;     V 

-6ab        '    \ 
12  ah) 

144  ELEMENTARY  ALGEBRA 

9/2 - ^f  1 + -2j_Y    X3.  f « + ^ Y+  f- - -T 

\       ^  +  yj\       ^-yj  \b     aj      \b     aj 

\5yj'  .  '   [Sb     3aJ  '^\4.b     3a 

\oy     4:Xj  \4:y       X  J      \4:y       X 

f3ab_2be\\  /^  .3?Y -/'?£- ^'' 

■   >^2c       9a;  ■   \5q^2pJ      \3q     2p 

21.   f5a^  +  l2a6-i^V— • 
V8  6'  16  aV     32  6* 

'.3(1  +  36     2a-36\  .   4a'-96' 
a  +  46y  ■  9a 


22. 


3a-46     3a  +  46y     9a2-166 


23    f   "  «'       I     .6  ^M  ■    2a6 

•   l^a  +  6      (a  +  5)2     a-6     a?-by'a?-b^' 

M        9W-r     32,/  ^g    J 4,6,.     9x. 

o^     (ia?     4(A      /2a^2c\  '   i^_ll        ?_-L 

^'-  V96^-^J-^(36  +  T;  6        9         ^     4x 


FRACTIONS  146 


33.    -f^—K\x    -'-'' 


4Va-6     a-\-bJ     a%-\-ab^     a^-\-b^ 
34.  ^  +  y  I  y  +  g  I  g  +  a; 

35.  ?-ri^^ ^ L_i 

a     L^a^^-l     l-2a     l  +  2aj 
x-2     raj-4      /2-3.T     2a;  +  Al 

a'b-a^b^-\-ab*-b'' 


38.    2 


[_a;_3      x-2] 


oc^  —  ox  +  Q 


112 


39.  1   '  1       .       1 

a;  —  -     1  +  -     1  — -2 
X  X  or 

40.   If  a  =  3,  6  =  4,  c  =  5,  d  =  2,  find  the  value  of 

a 


6  +  -i 


d 


Factor  the  following  expressions : 

41.    9-^  +  ^.  42.    ^-3^2. 

y     2/'  y^      y 

43.    ^'_8a|c     16a2  ^_J^  M^  +  l 

2/=^       by         b'  f     27  c^ 

46.  Simplify  l^x-^y -  (^x-iy)]^a{^x-iy). 

X ^y     ^_y 

47.  Simplify  Lj-g_g  +  ^^-y. 

^     ^      5  3     ^     15 

48.  Divide  i-^V^  +  ^x-^-A^^'  by  J  +  iaj-fo^. 


From 
it 

Simplif 

y  a^-y 

.  W-' .  a^-2-  .  a^+\ 

146  ELEMENTARY  ALGEBBA 

49. 

subtract 

50 

51.  A  man  left  half  his  property  to  his  wife,  one  third  to 
his  daughter,  and  the  remainder  to  his  son.  If  the  son  received 
$1700,  how  many  dollars  did  the  man  leave  ? 

52.  A  man  bought  five  pounds  of  coffee  of  two  different 
kinds  for  f  1.55.  If  the  better  kind  cost  35^  and  the  poorer 
kind  25  ^  per  pound,  how  many  pounds  did  he  buy  of  each  ? 

53.  A  and  B  set  out  walking  in  the  same  direction,  but  A 
has  a  start  of  2  miles.  A  walks  at  the  rate  of  3  miles,  and 
B  at  the  rate  of  3^  miles,  per  hour.  After  how  many  hours 
will  B  overtake  A,  and  how  far  will  he  have  walked  ? 

54.  The  sum  of  13  dollars  is  made  up  of  dollars,  half  dollars, 
and  quarters.  If  the  number  of  dollars  is  half  the  number  of 
quarters  and  four  times  the  number  of  half  dollars,  find  the 
number  of  each. 


CHAPTER   IX 

FRACTIONAL  AND  LITERAL  EQUATIONS 

FRACTIONAL  EQUATIONS 

163.  Clearing  of  fractions.  If  an  equation  contains  frac- 
tions, these  may  be  removed  by  multiplying  each  term  by  the 
L.  C.  M.  of  the  denominator. 

Ex.1.    Solve  ^-^^  =  12-^±i-aj. 
6  3  2 

Multiplying  each  term  by  6  (Axiom  3,  §  90), 

2x  -  2(x  -  3)=  72  -  3(a;  +  4)- 6aj. 
Removing  parentheses,  2x  —  2a;  +  6  =  72  —  3a;  —  12  —  6a;. 
Transposing,       2x  -  2a;  +  3x  +  6x  =  -  6  +  72  -  12. 
Uniting,  9  a;  =  54. 

x  =  Q. 
Check.     If  a;  =  6,  each  member  is  reduced  to  1. 

;  14  9 


Ex.  2.    Solve 


T:     x-\-1~      x-^3 
Multiplying  by,(x  -  1)  (x  +  1)  (a;  +  3), 

6(a;  +  l)(a;  +  3)  -14(a;  -l)(a;  +  3)  =  -  9  (x+  l)(a;  -  1). 

Simplifying,  5x^  +  20a;  +  15  -  14a;2  -  28a;  +  42  =  -  9a;2  +9. 
Transposing,         6a;2  -  14a;2  +  9a;2  +  20a;  -  28a;  =  -  15  -  42  +  9. 
Uniting,  _8a;  =  -  48, 

x  =  6. 
Check.    If  a;  =  6,  each  member  is  reduced  to  ~  1. 

147 


148  ELEMENTABY  ALGEBRA 


EXERCISE  66 


Solve  the  following  equations  : 
x-2     3-a; 


2. 


3  2 

a;  —  3     4:  —  x 


^     9a;-5  ,   .        2U-X 
3. |-4ic= 


^     13-a;  ,  ic     12 -a; 
'•    -3-  +  2  =  -^' 

5.    1^  +  .  =  41^. 

a;-2     x-\-2  _A  —  4:X 
'       2     "*"     6     ~     12     ' 

g;  — 3     cc  —  4_5  — a; 
*   ~^0~"^    15    ~    12   ' 

8.    ^^Ili  =  a,  +  4. 
o 

_     3x±W_2x±_^ 


16. 

x-3 

17. 

4  =  5-1. 

X        X 

18. 

a;  '  a;  '  a; 

19. 

4         7         1 
5x     10  a;     10 

20. 

a;      a;     3a; 

21. 

a;-2     3 
a;  +  3     4 

22. 

12H-a;     124-2a; 
2a;             3a; 

23. 

4a;-l      o     3^     Q 
5          ^  =  2"     ^- 

24= 

6a;  — 5      o     5a;      e- 
4         ^=  3      ^- 

10.  ^±^  =  3.  ■  25.    ^_l^+§^-^+f=101 
a;  — 3  2       3       4       5      6 

11.  ?«  =  13.  26.   £±3  =  ^. 

a;  X        a;  +  4 

12.  i5=i5.  27.   20-.  _.-6 


4a;  a;  — 8      15  — a; 

13.   M  +  7  =  12.  28.   =^  +  1-     '^ 


a; 


a;  — 4     a;  — 3 


14.  -65 ^13.  29.   2£±i  =  5. 

3a;  +  2  3a;-5     2 

15.  ^50__2^i3  3„     8^-30^  +  4^2. 
3s  +  l  12a^  +  5a;-3     3 


34. 
35. 
36. 


FBACTIONAL  AND  LITERAL  EQUATIONS  149 


31. 
32. 


3  4 

7(5fl;  +  2)      3(7  a;- 4)^  2(9  a;  + 2) 
6  5  5        * 


33.   l(2a;-9)+^(a:-4)  =  ^(3a:-2)-(3aj-5). 
lo  o  o 


8a;-7 

12  a;- 

-19 

6a;  +  7 

9a;- 

-2 

3 

2 

1 

a^  +  l 

a;-2 

a;  +  2 

'     4- 

2 

16 

x^2     x-2     x^-4: 


7  1  24 

37.    — h— ^— =  . 

a;  +  3     a;-3     ic^.g 

63  10 


42. 


43. 


44. 


38. 

7                 13              4 

2a;-5      6a;2_i5^     3^ 

39. 

15             20             10 

3a;-2     3a;2_2a.    3a;-2 

40. 

51        23  a; +26        22 
2a;+3      4a;2-9      2a;-3 

41. 

4             2         15a;+2 
3a;+2    3a;-2     9ar2-4 

66 

4a;2_2a;     6x2  +  3a;     4.x'-l 
4a;-7  ^    7a;  +  2        11 
6a;  +  18     10a;  +  30     45* 

19  5      ^         53a;  +  4 

3a;-2     5a;-6      (3  a; -2)  (5  a; -6)* 


^^5  7  5a;+l  ^^  2a;  +  5     a;4-29     -, 

45. =  ■  «  47.  — ~ — " =^  X. 

a;-7    a;+3    a.'2-4a;-21  a;-2       x  +  9 

^^7  3        26a;-25  ^^  6a;-ll      3a;-2      ^ 

4d. = •  4o. =  O. 

a;4-4    a;— 5    a.'^— x— 20  a;  — 11        aj  +  l 


49. 


5a;-l     2a;  +  9^14a;^  +  49 
2a;H-l     2a;-l       4a^-l  ' 


Sx-4r     2x-5^25x'-27x-42 
5x-6     5x-^6  2ox'^S6 

Qx-{-7     Sx-5     2a;2_35a.^_77 


51. 


2a;  +  3      3a;-4        6ar^  +  a;-12 


2a;-9      2a;-7  ^  (5  a;-13)(2  a;-4) 
*   3a;-l     2a;4-5  Sx'+lSx-d 


150  ELEMENTARY  ALGEBRA 

53.  ^-^-  =  1.  56.    ^_  +  ^_  =  3 

x  —  S     x  —  5     X  2a;  +  3     3ic— 1     x 

54.  ^ ?-  =  l  57.    ^-+       6  10 


a;  +  7     £c  — 5     ic  2a;  — 9     3a;— 1      Ax  — I 

55.    ^_  +  _l_  =  ^_  58.    A-        8  4  —  1 


a;  —  1      a;  —  5     x  —  4:  5x      10  a;  —  5      4ar'  —  1 

2a;-l         l-3a;-      2 


59. 


6a;^  —  4a;     9a;^4-6a;     3a; 


^^     5  a;-. 4  ,  1.3 -3a;      1.8  -  8  a; 
.3^2  1.2 

4(13a;-.6)      3(1.2 -a;)  ^9  a;  +  .2      54-llg? 
5  10  20  4       * 

62     2-2.8a;^2.1(.2+a;) 
2(a;-l)        .3 -1.5  a; 

164.   If  two   or  more   denominators   are  monomial,   and    the 

other  polynomial,  it  is  advisable  first  to  remove  the  monomial 
denominators  only,  and  after  simplifying  the  resulting  equa- 
tion to  clear  of  all  denominators. 

Ex.  1.    Solve  1^^±^  -  ?-^^  =  ^^l^. 
10  5a;-l  5 

Multiplying  each  term  by  10,  the  L.  CM.  of  the  monomial  denomina- 
tors, 

16a;  +  3-^^'^~^^  =  16a;-2. 
6a;  —  1 

Transposing  and  uniting,  5  =  ^  . 

5x  -  1 

Multiplying  by  5  a:  -  1,  25  a;  -  5  =  20  a;  -  50. 
Transposing  and  uniting,        5  a;  =  —  45. 
Dividing,  x  =  ^  9. 

Check.     Ux=  —  9,  each  member  is  reduced  to  —  -^. 


FRACTIONAL  AND  LITERAL  EQUATIONS  151 

165.   Frequently  it  is  advantageous  to  unite  terms  before  the 
clearing  of  fractions. 


Ex.2.    Solve  ^  +  1      a.  +  4_x  +  2     x  +  5 
x-{-2     x-^5     x-\-3     05  +  6 

Uniting  the  fractions  in  each  member  separately, 

x^  -^  Q  X  +  5  -  x^  -  6  X  -  S  _x^^  +  S X  -{-  12  -  x"^  -  Sx  ■ 

-16 

(x  +  2)(x  +  5)                           (x  +  3)(x+  6) 

» 

-3                         -3 

^'''  (a;4-2)(x  +  5)      (x  +  3)(x  +  6) 

Dividing  both  members  by  —  3  and  clearing  fractions, 

ic2  +  9x+  18  =  a;2  +  7a;  +  io. 
Transposing  and  uniting,      2  x  =  —  8. 
Dividing,  x  =  —  4. 

Check.   X  =  —  4  reduces  each  member  to  f  • 

Solve  the  following  equations : 


63. 


lOar  +  l      4.x-\-7  ^6a;-2 
5  6a;  +  ll  3 


64     21a;  +  7      7x-13^7x-^4: 
9  6a;  +  3  3 

24:X  +  7     8aj-fl      2a;-7 


65. 


66. 


15  5  7x-6 

17  a;- 13     34  a;- 75^13  a;  + 7 
14  28  6  a;  4- 14* 


lx-5     _x-l_     .  1  _  14  a;  -  3 

18a;  +  43      3a;  +  5  ^6a;     ^ 
15  2  a;- 25       5 

AQ     7a;  +  13  .  7a;-29     14^+19,  ii 


70. 


_6 6^4 

x—5     x—S     x—7 


73, 


152  ELEMENTARY  ALGEBRA 

71        1  1    _    1  1         Y2  ^— 1     a?— 4_a?— 5    a;— 8 

a;— 8     07—6     a;— 4     x—2  '  x—2     x—5    a;— 6     x—9 

x  —  4:     x  —  T__x~5      x  —  S 

x—5     x—S     x—6     x—d 
^4  x-{-2 x  +  U  6a;-4         ^g 

12a^  +  34a;  +  24     12a^-14a;-48     9aj2-12a;-32 

2-x     l-3a;       o  ,  §6 

1      ■      1  _5 5^ 

l_a;'^l  +  a;       2a;       2 

76.    (4aj-i)(5aj  +  f)  =  (2aj  +  i)(10aj-f). 

LITERAL  EQUATIONS 

166.  Literal  equations  (§  88)  are  solved  by  the  same  method 
as  numerical  equations. 

When  the  terms  containing  the  unknown  quantity  cannot  be 
actually  added,  they  are  united  by  factoring. 

Thus,  ax-\-bx  =  (a-{-  h)x. 

mx  +  rri^x  —  mux  =  (m-{-m^  —  nm)x, 

Ex.l.    Solve  E=^  +  £iiJ^_36  +  a. 

h  a  a 

Clearing  of  fractions,  ax  ~  a!^  -\-  hx  —  h^  =  —^l)^  —  ah. 
Transposing,  ax  +  hx  =  a"^  -{■  H^  —  3  6^  _  ah. 

Uniting,  ^  (^a +  h)x  =  a'^  -  ah -^h"^. 

Dividing,  ^^a^-ah-2hy 


a  +  6 
Reducing  to  lowest  terms,  cc  =  a  —  2  6. 

EXERCISE  67 

Solving  the  following  equations : 

1.  a4-a;  =  6  — a;.  3.    a+m+aj+9=2m+a+10. 

2.  a  — a;  =  6  — 8.  4.    aa;  =  6. 


FRACTIONAL  AND  LITERAL  EQUATIONS  153 

5.  mx  —  n=p. 

6.  a(x  —  b)=c. 

7.  a(b  —  xy=c. 

8.  4:(x-a)  =  Sx-5b. 

9.  S(4.a-Sx)  =  5(4:b-x). 

10.  mx  +  nx=p. 

11.  m  —  mx=px  —  d>  ax  —  b     a 

12.  aa; -f  &a;  =  ac  +  6.  25.    ^+-^  =  i 

13.  {a-\-b)x  =  m  —  cx. 

14.  (a  —  6)a;  =  2  a  —  (a  +  6)a; 

15.  -  =  6. 
a 


21 

a;  +  a      6      ic  — 6 

1  ^ 

baa 

'  b 

22. 

a{b  +  ^ym{c- 

■f> 

23. 

l-\-x_a 
1-x      b 

24. 

ax-\-b      c 

Wir 


16. 

17. 

a     b_ 

X     x~~ 

18. 

^-l  =  *-9. 

X                 X 

19. 

a  —  bx,^      bc  —  x 

— — —  +  6  = 

c                     c 

20. 

-  4-  T  =  c. 
a     b 

26. 

b  +  2x       ' 

27. 

a-^x     a-\-b 
a—x     a—b 

28. 

^         c-d 

b  +  x 

29. 

X   .   X   .   x        , 

a     b      c 

30. 

a  —  bm     c  —  bn 

=  1. 


mx  nx 


a—xb—x     c 
ol. 1 1- 


a  b  c 

32.    ^Zl^  +  £zi£  =  2. 
x—b     x—a 

x—b  ■  x  +  b     ax  —  b^_r. 
a  —  b     a-\-b      a^  —  b^ 

x-2a __  x-\-2b  ^ 3  a;- 3a 
a  +  6       2a-\-2b  2b 

35  11a;       ,  3  a(4  6  —  5  a)  _        a 

6a;  +  126        9a^-4:b^     ■~6a-4& 


34. 


154  ELEMENTARY  ALGEBRA 

a  b         a  —  b 


37. 


x  +  a     x-{-b     x  —  b 


38.    ^0>  —  ^)  I  b(c  —  a)^a-^b     /^,  «\ 
bx  ex  X         \G      b) 

5.629  +  - 

39     ^  a:  _  (5.629)^  +  1 

'    5.629-1      (^•629)^-1* 

X 

167.   It  frequently  occurs  that  the  unknown  letter  is  not 
expressed  by  x,  y,  or  z. 

Ex.   If  -^ 1  =  ^,^      ,  i  find  a  in  terms  of  b  and  c. 

3  a  — c       3(a  — c) 

Multiplying  by  3(«  -  c)(3  a  -  c) 

6(a  -  &)(a  -  c)  =  (2  a  +  6)(3  a  -  c). 

6  a2  -  6  a&  -  6  ac  +  6  6c  =  6  a2  -  2  ac  +  3  a6  -  6c. 

Transposing  all  terms  containing  a  to  one  member, 

-  6  a6  -  6  ac  +  2  ac  -  3  a6  =  -  6  6c  -  6c. 
Simplifying,  -^ ah  — 'kac^—l  he. 

Uniting  the  a,  and  multiplying  by  —  1,  a(9  6  +  4  c)  =  7  6c. 

Dividing,  a  =  -1^^. 

96  +  4C 

EXERCISE  68 

1.  If  -  =  -,  find  a  in  terms  of  b  and  c. 

0  z 

2.  If  7  =  c,  find  6  in  terms  of  a  and  c. 

3.  If  __±L^  =-  _   solve  for  c. 

1  — a     c 

4    If =  -,  find  a  in  terms  of  b  and  a 

1  — a     c 


FRACTIONAL  ANB  LITERAL  EQUATIONS  155 

6.    -=-  +  -,  solve  for/. 
f     P     Q 

6.  Solve  the  same  equation  for  p. 

7.  The  formula  for  simple  interest  (§  30,  Ex.  3)  is  {  =  ^, 

i  denoting  the  interest,  p  the  principal,  r  the  number  of  %,  and 
n  the  number  of  years.     Find  the  formula  for : 

(a)  The  principal. 

(b)  The  rate. 

(c)  The  time,  in  terms  of  other  quantities. 

8.  The  formula  for  falling  bodies  (§  30,  Ex.  7)  is  s=i  gf, 
Find: 

(a)  g  in  terms  of  s  and  t. 

(b)  f  in  terms  of  a  and  s. 

9.  (a)  Find  a  formula  expressing  degrees  of  Fahrenheit  (F) 
in  terms  of  degrees  of  centigrade  ((7)  by  solving  the  equation 

0  =  |(2^- 32). 

(6)  Express  in  degrees  Fahrenheit  40°  C,  100°  C,  -20°C. 

10.  The  formula  for  compound  interest  is  I=p{  1 + -rzr: )  —p- 
Express  p  in  terms  of  /,  ?*,  and  n.  ^  ^ 

11.  If  C  is  the  circumference  of  a  circle  whose  radius  is  i?, 
then  C  =  2  irR.     Find  R  in  terms  of  C  and  tt. 

12.  If  a^  =  ft'*  +  c^  —  2  cp,  find  p  in  terms  of  a,  6,  and  c 

13.  ^  =  _^,  solve  for  a?. 
h     c—  X 

14.  F=l(b  +  c).     Find: 

(a)  ^  in  terms  of  F,  6,  and  c. 
(6)  6  in  terms  of  F,  h,  and  c. 

15.  If  0=     ^    ,  find  r  in  terms  of  (7,  E,  and  J?. 


156  ELEMENTARY  ALGEBRA 


IDENTICAL  EQUATIONS 

168.  Identical  equations  may  be  proved  by  applying  the  same 
operations  which  are  used  in  the  solution  of  equations.  If  the 
resulting  equations  prove  to  be  true,  then  the  original  equation 
is  true. 

Care,  however,  must  be  taken  not  to  multiply  the  two  mem- 
bers of  an  equation  by  zero,  since  the  axiom  of  dividing  equals 
by  equals  cannot  be  applied  to  zero  divisors. 

Ex.  1.    Prove  that — ^  =  —^ 

a      Zoo       2  ah      h 

The  identity  is  true, 
if  26—  (a  +  6)=«  +  &  —  2a  (clearing  of  fractions), 

or,  if  h  —  a  =  h  —  a  (simplifying). 

But  since  6  —  a  =  6  —  a, 

1  _  a-\-h  _  a  +  h  __  1 
a      2  ah        2  ah       h 

169.  Frequently  a  relation  between  letters  is  given  in  the 
form  of  an  equation,  and  another  equation  has  to  be  proved. 

Ex.  2.    If  ad  =  he,  prove  that  ^^^  =  ^i^. 

a  —  h     c  —  d 

The  identity  ^^  =  ^-^  is  true, 
a  —  b     c  —  d 

if  (a  +  &) (c  —  d)  =  (a  —  6)  (c  4-  d)  (clearing  of  fractions), 

or,  if  ac+hc  —  ad  —  bd  =  ac  —  bc  +  ad—  hd  (simplifying), 
or,  if  be  —  ad  =—hc-\-  ad  (canceling), 

or,  if  2  he  =  2  ad  (transposing), 

or,  if  hc  =  ad  (dividing).^ 

But  he  =  ad. 


Hence 


a-\-  b  _c-\-  d 
a  —  b     e  —  d 


FRACTIONAL  AND  LITERAL  EQUATIONS  157 

BXEBCISE  69 
Prove  the  following  identities : 

1     ^  +  ^     b~a_b^  +  2ab-a* 


2, 


a  b  ab 

2oi^ y__  _     X        x  —  y_ 

a^  — /     x-\-y     x  —  y     x-\-y 

2a+3  g+l     ^       a+2 

o?-la-Q     a?-a-%     a" -2  a -3 
2a-3  5      ^3      2a-7 

4a^  — a     2a  — 1      a     4a^  — 1 

a^      .      6  a  62 


a'-b'     a  +  b     a-b     a'-b^ 
If  ad  =  be,  prove  that :  If  ac  =  6^,  prove  that : 

6.  ^=:^.  15.    ^  =  5. 

c      d  be 

„    a^b_c+d 

lb. 


8. 


b  d 

a  —  b  _c  —  d 

a— c     6— d 


17. 


a         a 

-b 

aH-  6     a- 
a'  +  2b' 

—  c 
62 

— "  =  -T —  If  a^  +  6^  =  c^,  prove  that : 

10    «4-26^c  +  2(?_  18-  (a-6/  +  2a6=c2. 

•  a  +  36     c  +  Sd  19.  (a  +  6)2+ (a- 6)2  =  2c*, 

2a  — 6        2c--d 

*  ap  +  6g       cp  +  dq  '      '^[fij      \ij' 
-«    a  +  26  — c— 2c2     c-^2d  h               n 

lo.     :: z = r •  _             1  _)_  _ 


t)<. 


b-^d  d 


22. 


c  c 


^4    3a+66-5c-10d^a4-26  ^_a        6 

36— 6d  6  c         c 


168  ELEMENTARY  ALGEBRA 

PROBLEMS  LEADING  TO  FRACTIONAL  AND   LITERAL 
EQUATIONS 

170.  Ex.  1.  When  between  3  and  4  o'clock  are  the  hands  of 
a  clock  together  ? 

At  3  o'clock  the  hour  hand  is  15  minute  spaces  ahead  of  the  minute 
hand,  hence  the  question  would  be  formulated  :  After  how  many  minutes 
has  the  minute  hand  moved  15  spaces  more  than  the  hour  hand  ? 

Let  -  X  =  the  required  number  of  minutes  after  3  o'clock, 

then  X  =  the  number  of  minute  spaces  the  minute  hand 

moves  over, 
and  ^  =  the  number  of  minute  spaces  the  hour  hand  moves 

over, 
the  number  of    minute  spaces  the  minute  hand 
moves  more  than  the  hour  hand. 


Or 


12 


Uniting,  ii^  =  15. 

Multiplying  by  12,  11  a;  =  180. 

Dividing,  x  =  ^t^-  =  16/,-  minutes  after  3  o'clock. 

Ex.  2.  A  can  do  a  piece  of  work  in  3  days  and  B  in  2  days. 
In  how  many  days  can  both  do  it  working  together  ? 

If  we  denote  the  required  number  of  days  by  x  and  the  piece  of  work 
by  1,  then  A  would  do  each  day  \  and  B  ^,  while  in  x  days  they  would  do 

respectively  -  and  -,  and  hence  the  sentence  written  in  algebraic  symbols 

3  2 

is^  +  ^  =  l. 

3      2 

A  more  symmetrical  but  very  similar  equation  is  obtained  by  writing 
in  symbols  the  following  sentence  :  "  The  work  done  by  A  in  one  day  plus 
the  work  done  by  B  in  one  day  equals  the  work  done  by  both  in  one  day." 

Let  X  =  the  required  number  of  days. 

Then  -  =  the  part  of  the  work  both  do  in  one  day. 

X 

Therefore,  1  +  1  =  1. 

'  3     2     X 

Solving,  x  =  h  or  1  J,  the  required  number  of  days. 


FRACTIONAL  AND  LITERAL   EQUATIONS 


159 


Ex.  3.  The  speed  of  an  express  train  is  f  of  the  speed  of  an 
accommodation  train.  If  the  accommodation  train  needs  4 
hours  more  than  the  express  train  to  travel  180  miles,  what  is 
the  rate  of  the  express  train  ? 


Time 
(hours) 

Kate 
(miles  per  hour) 

Distance 
(miles) 

Express  train 

5         X 

9x 
6 

180 

Accommodation  train 

I80^^  =  il« 

X 

X 

180 

Therefore, 

Clearing, 

Transposing, 

Hence 

Explanation : 


180^100     ^ 

X  X 

180  =  100  +  4  X. 
4a;  =  80. 
a;  =  20. 
I  a;  =  36  =  rate  of  express  train. 


(1) 


9a; 


If  X  is  the  rate  of  the  accommodation  train,  then  ^^^  is 

6 


the  rate  of  the  express  train.    But  in  uniform  motion,  Time 


Distance 
Kate 


Hence  the  rates  can  be  expressed,  and  the  statement,  "The  accommo- 
dation train  needs  4  hours  more  than  the  express  train,"  gives  the  equa- 
tion (1). 

EXERCISE  70 

1.  Find  a  number  whose  third,  fourth,  and  fifth  parts  added 
together  make  47. 

2.  Find  a  number  whose  third  part  exceeds  its  fourth  part 
by  4. 

3.  What  number  exceeds  the  difference  of  its  third  and 
fifth  parts  by  26? 

4.  Two  numbers  differ  by  5,  and  one-half  the  greater  exceeds 
the  smaller  by  1.     Find  the  numbers. 


160  ELEMENTARY  ALGEBRA 

6.  Two  numbers  differ  by  9,  and  -^-^  of  the  greater  is  equal 
to  -f  of  the  smaller.     Find  the  numbers. 

6.  Find  two  consecutive  numbers  such  that  \  of  the  greater 
exceeds  ^  of  the  smaller  by  1. 

7.  The  sum  of  two  numbers  is  56,  and  one  is  f  of  the  other. 
What  are  the  numbers  ? 

8.  A  certain  number  is  increased  by  2,  the  sum  multiplied 
by  3,  the  product  divided  by  2,  the  quotient  increased  by  2,  the 
sum  multiplied  by  f,  and  the  result  is  found  to  be  12.  What 
is  the  number  ? 

9.  Find  the  sum  of  three  consecutive  numbers  such  that 
one  half  of  the  first,  plus  one  third  of  the  second,  plus  one 
fourth  of  the  third,  equals  16. 

10.  A's  age  is  one  third  of  his  age  30  years  hence ;  how  old 
is  he? 

11.  Five  years  ago  a  boy's  age  was  one  third  of  his  age  six 
years  hence.     How  old  is  he  now  ? 

12.  A  boy's  age  is  one  fourth  of  his  father's  age,  and  in  five 
years  his  age  will  be  one  third  of  his  father's  age.  Find  the 
father's  present  age. 

13.  A  is  10  years  older  than  B,  and  i  of  B's  age  is  equal  to 
-|  of  A's  age.     Find  their  ages. 

14.  A  is  40  years  old,  and  B  is  half  as  old  as  A.  In  how 
many  years  will  B  be  y\  as  old  as  A? 

15.  The  sum  of  the  ages  of  a  father  and  his  son  is  36,  and 
in  10  years  the  son's  age  will  be  f  of  the  father's  age.  Find 
their  ages. 

16.  A  post  is  a  third  of  its  length  in  the  ground,  a  fourth  of 
its  length  in  water,  and  14  feet  above  w5,ter.  What  is  the 
length  of  the  post  ? 

17.  A  man  left  $  41,000  to  his  wife,  two  sons,  and  two  daugh- 
ters. Each  son  received  f ,  and  each  daughter  |,  of  the  wife's 
share.     What  was  the  share  of  each  ? 


FRACTIONAL  AND  LITERAL  EQUATIONS  161 

18.  A  man  left  ^  of  his  property  to  his  wife,  ^  to  his  daugh- 
ter, and  the  remainder,  which  was  $  14,000,  to  his  son.  How 
much  money  did  the  man  leave  ? 

19.  A  man  lost  |  of  his  fortune  and  $  1000,  and  found  that 
he  had  left  ^  of  his  original  fortune.  How  much  money  had 
he  at  first  ? 

20.  A  and  B  together  have  $  500.  If  A  gains  $  150,  and  B 
loses  $  100,  B's  money  will  be  f  of  A's  money.  How  much 
has  each? 

21.  After  expending  one  third  of  his  money  and  $20,  a 
man  had  left  one  fourth  of  his  money  and  $  30.  How  much 
money  had  he  at  first  ? 

22.  A  train  traveling  30  miles  per  hour  starts  f  of  an  hour 
before  a  second  train  that  travels  35  miles  an  hour.  In  how 
many  hours  will  the  first  train  be  overtaken  by  the  second  ? 

23.  An  express  train  starts  from  a  certain  station  three 
hours  after  an  accommodation  train,  and  after  traveling  160 
miles  overtakes  the  accommodation  train.  If  the  rate  of  the 
express  train  is  |  of  the  rate  of  the  accommodation  train,  what 
is  the  rate  of  the  latter?    (§  170,  Ex.  3.) 

24.  The  length  of  a  rectangle  exceeds  the  width  by  5^  feet. 
If  the  length  were  decreased  by  3  feet  and  the  width  increased 
by  1^  feet,  the  area  would  not  be  altered.    Find  the  dimensions. 

25.  If  each  side  of  a  square  were  increased  by  1^  feet,  the 
area  would  be  increased  by  21f  square  yards.  Find  the  side 
of  the  square. 

26.  The  width  of  a  room  is  f  of  its  length.  Another  room 
is  10  feet  shorter  and  1  foot  wider.  If  the  second  room  con- 
tains 150  square  feet  less  than  the  first  room,  find  the  dimen- 
sions of  the  room. 

27.  If  the  radius  of  a  circle  were  increased  by  one  yard,  its 
area  would  be  increased  by  66  square  yards.  If  it  is  assumed 
as  3|,  find  the  radius  of  the  circle.  (The  area  s  of  a  circle 
whose  radius  is  R,  is  found  by  the  formula  s  =  irE^.) 


162  ELEMENTABY  ALGEBRA 

28.  When  between  one  and  two  o'clock  are  the  hands  of  a 
watch  together  ?     (§  170,  Ex.  1.) 

29.  At  what  time  between  2  and  3  o'clock  are  the  hands  of 
a  clock  together  ? 

30.  At  what  time  between  3  and  4  o'clock  are  the  hands  of 
a  clock  in  a  straight  line  and  opposite  ? 

31.  At  what  time  between  6  and  7  o'clock  are  the  hands  for 
the  first  time  at  right  angles? 

32.  At  what  time  between  6  and  7  o'clock  are  the  hands  for 
the  second  time  at  right  angles  ? 

33.  A  has  invested  capital  at  4|^%  and  B  has  invested 
$10,000  more  than  A  at  4%.  They  both  derive  the  same 
income  from  their  investments.  How  much  money  has  each 
invested  ? 

34.  A  man  has  \  of  his  money  invested  at  4%,  one  third  at 
3|^%,  and  the  remainder  at  5%.  How  much  money  has  he 
invested  if  his  annual  interest  therefrom  is  $12,000? 

35.  A  man  has  a  certain  sum  of  money  invested  at  5%,  and 
twice  that  amount  at  4%.  The  sum  invested  at  4%  brings 
$350  more  interest  than  the  one  invested  at  5%.  How  much 
money  is  invested  at  4  %  ? 

36.  What  principal  invested  at  4  %  simple  interest  for  two 
years  will  amount  to  $  27,000  ? 

37.  Divide  65  into  two  parts,  such  that,  if  the  greater  is 
divided  by  the  smaller,  the  quotient  is  f . 

38.  The  sum  of  two  numbers  is  123,  and  if  the  greater  is 
divided  by  the  smaller,  the  quotient  is  2  and  the  remainder  is 
3.     Find  the  numbers. 

39.  An  ounce  of  gold  when  weighed  in  water  loses  rf^  of  an 
ounce,  and  an  ounce  of  silver  2t  ^^  ^^  ounce.  How  many 
ounces  of  gold  and  silver  are  there  in  a  mixed  mass  weighing 
100  ounces  in  air,  and  losing  7  ounces  when  weighed  in  water  ? 


FB ACTIONAL  AND  LITERAL  EQUATIONS  163 

40.  A  can  do  a  piece  of  work  in  2  days,  and  B  in  4  days. 
In  how  many  days  can  both  do  it  working  together  ?  (§  170, 
Ex.  2.) 

41.  A  can  do  a  piece  of  work  in  3  days,  and  B  in  4  days. 
In  how  many  days  can  both  do  it  working  together  ? 

42.  A  can  do  a  piece  of  work  in  4  days,  and  B  in  5  days. 
In  how  many  days  can  both  do  it  working  together  ? 


171.  The  last  three  questions  and  their  solutions  differ  only 
in  the  numerical  values  of  the  two  given  numbers.  Hence,  by 
taking  for  these  numerical  values  two  general  algebraic 
numbers,  e.g.  m  and  n,  it  is  possible  to  solve  all  examples  of 
this  type  by  one  example.  Answers  to  numerical  questions  of 
this  kind  may  then  be  found  by  numerical  substitution.  The 
problem  to  be  solved,  therefore,  is : 

A  can  do  a  piece  of  work  in  m  days  and  B  in  w  days.  In 
how  many  days  can  both  do  it  working  together? 

If  we  let  X  =  the  required  number  of  days,  and  apply  the 

method  of  §  170,  Ex.  2,  we  obtain  the  equation  — 1_  _  =  _. 
mn  m     n     X 

Solving,  X  —  — ; — 

Therefore  both  working  together  can  do  it  in  — ~  days. 

To  find  the  numerical  answer,  if  A  can  do  this  work  in  6 

n     o 

days  and  B  in  3  days,  make  m  =  6  and  n  =  3.     Then  =  2, 

i.e.  they  can  both  do  it  in  2  days. 


Solve  the  following  problems  : 

43.   In  how  many  days  can  A  and  B  working  together  do  a 
piece  of  work  if  each  alone  can  do  it  in  the  following  number 

^^d^^s-  (a)  A  in  6,  Bin  6. 

(6)  A  in  12,  B  in  4. 

(c)  A  in  12,  B  in  6. 

(d)  A  in  20,  B  in  5. 


164  ELEMENTARY  ALGEBRA 

44.  Find  three  consecutive  numbers  whose  sum  equals  21. 

45.  Find  three  consecutive  numbers  whose  sum  equals  135. 

The  last  two  examples  are  special  cases  of  the  following 
problem  : 

46.  Find  three  consecutive  numbers  whose  sum  equals  m. 
Find  the  numbers  if  m  =  24;  30,009;  918,414. 

47.  Find  two  consecutive  numbers  the  difference  of  whose 
squares  is  17. 

48.  If  the  sides  of  a  square  were  increased  1  yard,  the  area 
would  be  increased  by  23  square  yards.  Find  the  side  of  the 
square. 

49.  A  battalion  of  soldiers  is  drawn  up  in  the  form  of  a  solid 
square.  To  form  a  solid  square  containing  in  each  side  one 
soldier  more,  27  more  men  are  required.  How  many  men  are 
there  in  the  square  ? 

The  last  three  examples  are  special  cases  of  the  following  one : 

50.  The  difference  of  the  squares  of  two  consecutive  numbers 
is  m ;  find  the  smaller  number.  By  using  the  result  of  this 
problem,  solve  the  following  ones : 

I.    Find  two  consecutive  numbers,  the  difference  of  whose 
squares  is  (a)  47,  (6)  121,  (c)  1517,  (d)  10,475,429. 

II.  If  the  sides  of  a  square  were  increased  by  one  foot,  the 
area  would  be  increased  (a)  17,  (h)  917,  (c)  415,673  square 
feet.     Find  the  side  of  the  square. 

51.  The  sum  of  the  three  angles  of  any  triangle  is  equal  to 
180°.  In  a  given  triangle  the  second  angle  equals  three  times  the 
first,  and  the  third  angle  5  times  the  first.    Find  the  first  angle. 

52.  The  second  angle  of  a  triangle  equals  m  times  the  first, 
and  the  third  equals  n  times  the  first.     Find  the  first  angle. 

Solve  the  problem  if : 

(a)  the  second  angle  =  5  times  the  first,  the  third  angle  =  3 
times  the  first. 


FRACTIONAL  AND  LITERAL   EQUATIONS  165 

(b)  the  second  angle  =  2  times  the  first,  the  third  angle  =  9 
times  the  first. 

(c)  the  second  angle  =  7|  times  the  first,  the  third  angle  =  6J 
times  the  first. 

53.  If  the  radius  of  a  circle  is  increased  by  1  foot,  the  area 
is  increased  by  m  square  feet.     Find  the  radius  of  the  circle. 

Solve  the  problem  if,  by  increasing  the  radius  by  1  foot, 
the  area  is  increased  (a)  by  44,  (6)  11,  (c)  286  square  feet. 

(  Assume  7r  =  — ) 

64.  Two  men  start  at  the  same  hour  from  two  towns,  100 
miles  apart,  the  first  traveling  3^  miles  per  hour,  and  the  sec- 
ond 6^  miles  per  hour.  After  how  many  hours  do  they  meet, 
and  how  many  miles  does  each  travel  ? 

55.  Two  men  start  at  the  same  time  from  two  towns,  d  miles 
apart,  the  first  traveling  at  the  rate  of  m,  the  second  at  the 
rate  of  n  miles  per  hour.  After  how  many  hours  do  they 
meet,  and  how  many  miles  does  each  travel  ? 

Solve  the  problem  if  the  distance,  the  rate  of  the  first,  and 
the  rate  of  the  second,  are  respectively : 

(a)  48  miles,    2  miles  per  hour,    4  miles  per  hour. 

(b)  25  miles,    3  miles  per  hour,    7  miles  per  hour. 

(c)  18  miles,  2^  miles  per  hour,  3^  miles  per  hour. 

56.  In  how  many  minutes  will  the  minute  hand  of  a  clock 
move  over  n  minute  spaces  more  than  the  hour  hand  ? 

Note.  The  greater  number  of  examples  relating  to  movements  of  the 
hands  of  a  clock  can  be  solved  by  determining  from  the  data  of  the 
exaiBple  the  number  of  minute  spaces  the  minute  has  to  move  more  than 
the  hour  hand,  and  by  substituting  this  number  for  n  in  the  answer  of  the 
last  example. 

57.  When  are  the  hands  of  a  clock  for  the  first  time  together 
after  (a)  1  o'clock,  (b)  6  o'clock,  (c)  9  o'clock  ? 

58.  When  are  the  hands  of  a  clock  opposite,  and  in  the  same 
straight  line  after  (a)  2  o'clock,  (b)  4  o'clock,  (c)  8  o'clock  ? 


166  ELEMENTARY  ALGEBRA 

59.  A  cistern  can  be  filled  by  two  pipes  in  m  and  n  minutes 
respectively.  In  how  many  minutes  can  it  be  filled  by  the  two 
pipes  together?  Find  the  numerical  answer,  if  m  and  n  are 
respectively  (a)  4  and  5  minutes,  (5)  7  and  42  minutes,  (c)  2 
and  3  hours. 

60.  A  cistern  can  be  filled  by  three  pipes  in  3,  4,  and  5  hours 
respectively.  In  how  many  hours  can  it  be  filled  by  all  three 
together  ? 

61.  State  and  solve  a  literal  problem  which  comprises  all 
examples  of  the  type  of  the  preceding  one.  Find  the  answer 
if  the  pipes  fill  the  cistern  respectively  in  (a)  2,  3,  4  hours,  (b) 
6,  30,  and  20  minutes. 

62.  A  cistern  can  be  filled  by  two  pipes  in  3  and  12  hours 
respectively,  and  emptied  by  a  third  pipe  in  5  hours.  In  how 
many  hours  can  the  cistern  be  filled  if  all  the  pipes  are  run- 
ning together?  Can  the  last  answer  be  obtained  from  the 
general  answer  of  the  preceding  one? 

63.  A  cistern  has  3  pipes.  The  first  will  fill  the  cistern 
in  7  minutes,  the  second  in  42  minutes,  and  the  three  running 
together  in  5  minutes.  In  how  many  minutes  will  the  last  one 
fill  it? 


CHAPTER  X 

RATIO   AND   PROPORTION 

RATIO 

172.  The  ratio  of  two  numbers  is  the  quotient  obtained  by 
dividing  the  first  number  by  the  second. 

Thus  the  ratio  of  a  and  ft  is  -  or  a  -f-  6.     The  ratio  is  also  frequently 
h 
written  a  :  6,  the  symbol  :  being  a  sign  of  division.     (In  most  European 
countries  this  symbol  is  employed  as  the  usual  sign  of  division.)     The 
ratio  of  12 :  3  equals  4,  6 :  12  =  .5,  etc. 

173.  A  ratio  is  used  to  compare  the  magnitude  of  two 
numbers. 

Thus,  instead  of  writing  "a  is  6  times  as  large  as  6,"  we  may  write 
a :  &  =  5. 

174.  The  first  term  of  a  ratio  is  the  antecedent,  the  second 
term  the  consequent. 

In  the  ratio  a :  &,  a  is  the  antecedent,  6  is  the  consequent.  The  numera- 
tor of  any  fraction  is  the  antecedent,  the  denominator  the  consequent. 

175.  The  ratio  -  is  the  inverse  of  the  ratio  — 

a  b 

176.  Since  a  ratio  is  a  fraction,  all  ^jrmcipZes  relating  to 
fractions  may  he  applied  to  ratios.  E.g.  a  ratio  is  not  changed 
if  its  terms  are  multiplied  or  divided  by  the  same  number,  etc. 

Ex.  1.   Simplify  the  ratio  2^ :  3J. 

2^:3i  =  f:^5Q=^XT^  =  f  =  3:4. 

A  somewhat  shorter  way  would  be  to  multiply  each  term  by  6. 

167 


168  ELEMENTARY  ALGEBRA 

Ex.  2.    Transform  the  ratio  5  :  3f  so  that  the  first  term  will 
equal  1.  03 

Ex.  3.    Solve  the  equation  a  :  a?  =  9. 

X 

Clearing,  a  =  9  a;. 

Dividing,  cc  =  -• 

EXERCISE  71 
Find  the  value  of  the  following  ratios : 

1.  144:36.  5.    $6.00:  $2.00. 

2.  5f  :  6|.    .  6.    If  hours  :  36  seconds. 

3.  |:i  7.    6  yards  :  12  feet. 

4.  16f  :  If.  8.    6  dollars  :  75  cents. 

Simplify  the  following  ratios  : 

9.   2^:3^.  14.  49  cc^^/ :  63  a;.yl 

10.   18f:26|.  15.  4.{a^h)x'.^{a-\-h)y. 

n.   41:3.  16.  (x^-l)'.{x  +  l)\ 

^  17.  (a;=^-3a;4-2):(a;2_4a;+4). 

13.    llxy'.^xy.  '   a;  — I'a;^  — 1 

Transform  the  following  ratios  so  that  the  antecedents  equal 
one: 

19.  3a2:15al  21.   4:12. 

20.  3|:4f  22.    5:12. 

Solve  the  following. equations : 

23.  26:ic  =  13.  25.   a:x  =  h. 

24.  45:4aj  =  15.  26.    63:7aj  =  9. 


RATIO  AND  PROPORTION  169 

27.  5abc:4:X=15bc.  30.   b:x  =  a  —  c. 

28.  16:(a;^3)  =  4.  31.    150  :  (3 a;  + 1)  =  H. 

29.  144:(2a;-hl)  =  16. 

32.  If  150  ounces  of  gold  cost  $3060,  and  20  ounces  of  silver 
$  12 J  find  the  ratio  of  the  value  of  gold  to  the  value  of  silver. 

PROPORTION 

177.  A  proportion  is  a  statement  expressing  the  equality  of 
two  ratios. 

f  =  fora:6  =  c:(Z  are  proportions. . 

178.  The  first  and  fourth  terms  of  a  proportion  are  the 
extremes,  the  second  and  third  terms  are  the  means.  The  last 
term  is  the  fourth  proportional  to  the  first  three. 

In  the  proportion  a:b  =  c:d,  a  and  d  are  the  extremes,  b  and  c  the 
means.    The  last  term  d  is  the  fourth  proportional  to  a,  &,  and  c. 

179.  If  the  means  of  a  proportion  are  equal,  either  mean 
is  the  mean  proportional  between  the  first  and  the  last  terms, 
and  the  last  term  the  third  proportional  to  the  first  and  second 
terms. 

In  the  proportion  a  :  &  =  6  :  c,  &  is  the  mean  proportional  between  a 
and  c,  and  c  is  the  third  proportional  to  a  and  b. 

180.  Quantities  of  one  kind  are  said  to  be  directly  proportional 
to  quantities  of  another  kind,  if  the  ratio  of  any  two  of  the 
first  kind  is  equal  to  the  ratio  of  the  corresponding  two  of  the 
other  kind. 

If  4  ccni.  of  iron  weigh  80  grams,  then  6  ccm.  of  iron  weigh  45  grams, 
or  4  ccm.  :  6  ccm,  =  30  grams  :  45  grams.  Hence  the  weight  of  a  mass  of 
iron  is  proportional  to  its  volume. 

Note.  Instead  of  "directly  proportional"  we  may  say  briefly  "pro- 
portional." 


170  ELEMENTARY  ALGEBRA 

Quantities  of  one  kind  are  said  to  be  inversely  proportional  to 
quantities  of  another  kind,  if  the  ratio  of  any  two  of  the  first 
kind  is  equal  to  the  inverse  ratio  of  the  corresponding  two  of 
the  other  kind. 

If  6  men  can  do  a  piece  of  work  in  4  days,  then  8  men  can  do  it  in 
3  days,  or  6  :  8  equals  the  inverse  ratio  of  4  :  3,  i.e.  3  : 4.  Hence  the  num- 
ber of  men  required  to  do  some  work,  and  the  time  necessary  to  do  it,  are 
inversely  proportional. 

181.  In  any  proportion  the  product  of  the  means  is  equal  to  the 
product  of  the  extremes. 

Let  a:b  —  c:d, 

a     c 

or  -z=  — 

b     d 
Clearing  of  fractions,        ad  =  be. 

182.  TJie  mean  proportional  between  two  numbers  is  equal  to 
the  square  root  of  their  product. 

Let  the  proportion  he     a:b  =  b  :  c. 

Then  b'  =  ac.     (§181.) 

Hence  b  =  ^s/ac. 

183.  If  the  product  of  two  numbers  is  equal  to  the  product  of 
two  other  numbers,  either  pair  may  be  made  the  means,  and  the 
other  pair  the  extremes,  of  a  proportion.     (Converse  of  §  181.) 

If  mn  =  pq,  and  we  divide  both  members  by  nq,  we  have 

—  —  t. 
q      n 

Ex.1.   Finda;,  if  6:a;  =  12:7. 

12  a;  =  42.     (§181.) 
Hence  a;  =  f  f  =  3J. 

Ex.  2.  Determine  whether  the  following  proportion  is  correct 
or  not :  8  :  5  =  7  :  4|. 

8  X  4|  =  35,  and  6  x  7  =  35 ;  hence  the  proportion  is  correct. 


RATIO  AND  PROPORTION  171 

184L  li  a:h=  c:d,  then 

I.  h:a  =  d:c.     (Frequently  called  Inversion.) 

II.  a\c  —  h:d.     (Called  Alternation.) 

III.  a-\-h:hz=c-\-d'.d.     (Composition.) 
Or  a  +  6  :  a  =  c  +  d  :  c. 

IV.  a  —  b:b  =  c  —  d:d.     (Division.) 

V.   a-{-b:a  —  b  =  c-{-d:c  —  d.     (Composition  and  Division.) 

Any  of  these  transformations  may  be  proved  by  the  method 

of  §  169,  although  in  many  cases  shorter  proofs  exist. 

m  a  —  bc  —  d 

To  prove,  e.g.  —_=—_. 

o  d 

This  is  true  if  ad  —  bd  =  hc  —  hd. 
Or  if  ad  =  be. 

But  ad  =  be.     (§  181.) 


Hence 


a  —  b     c  —  d 


lb  d 

185.   These  transformations  are  used  to  simplify  proportions. 

I.  Change  the  proportion  4 :  5  =  a; :  6  so  that  x  becomes  the 
last  term. 

By  inversion  5  :  4  =  6  :  a;. 

II.  Alternation  shows  that  a  proportion  is  not  altered  when 
its  antecedents  or  its  consequents  are  multiplied  or  divided  by 
the  same  number. 

E.g.  to  simplify  48  :  21  =  32  : 7  a;,  divide  the  antecedents  by  16,  the 
consequents  by  7,  3  :  3  =  2  :  a;. 

Or  1:1  =  2  :a;,  I.e.  a;  =  2. 

III.  To  simplify  the  proportion  5  :  6  =  4  —  a; :  a;. 
Apply  composition,  11  : 6  =  4  :  a;. 

IV.  To  simplify  the  proportion  8  :  3  =  5  +  a; :  a?. 

Apply  division,  5  :  3  =  5  :  x. 

Divide  the  antecedents  by  5,  1 :  3  =  1  :  aj. 


172  ELEMENTARY  ALGEBRA 

m-|-3  n     m  +  x 


V.   To  simplify 


m  —  3  71     m  —  x 


Apply  composition  and  division,  — ^  =  — ^ 

6n      2x 

Or  ^  =  ^. 

3  w      a; 

Dividing  the  antecedents  by  w,      —  =  -• 

3?i     a; 

Note.    A  parenthesis  is  understood  about  each  term  of  a  proportion. 


EXERCISE  72 
Determine  whether  the  following  proportions  are  true  or  not : 

1.  6i:2|-  =  8:3.  4.    7  :  10  =  5  :  7f 

2.  10:5i  =  4:2f  5.    Sa:  5h' =  5a:3^b\ 

3.  9:25  =  4:111 

Simplify  the  following  proportions  and  determine  whether 
they  are  true  or  not : 

6.  240:81  =  95:9,  8.    .65  :  .91  =  .4  :  .56. 

7.  57:69  =  38:46.  9.    ISJ  :  11^  =  6f  :  5|. 

^       10.    lli:12f  =  l|:2/T. 

11.  (a'-b^:(a  +  by=:(a-\-b):(a-b). 

12.  a^  —  7f:x^-]-xy  +  y^  =  x^  —  y^:x-\-y. 

Simplify  the  following  proportions  and  determine  the  value 
of  x: 

13.  12:16  =  15:a;.  18.  96  :  72  =  4aj :  21. 

14.  28:  35  =  16:  a;.  19.  135  :  5a;  =  72  :  64. 

15.  25:  55  =  35:  a;.  20.  .36  :  .8  a;  =  .105  :  .63. 

16.  48:  75  =  a;:  32.  21.  6| :  26J  =  a; :  2^. 

17.  a;:  38  =  15: 19.  22.  ^ab  :  ^bc  =  x\^-^cd. 


RATIO  AND  PROPORTION  178 

23.    (a  +  6):(a^6)  =  a::g-i). 

Find  the  fourth  proportional  to : 

25.  1,  2,  3.  27.   2,  4^,  9J.  29.   a^,  a6,  ac. 

26.  2,  3,  4.  28.    a,  6,  c.  30.    m^,  mn,  mn\ 

Find  the  third  proportional  to : 

31.  9  and  6..  33.   25  and  15.  85.   1  and  m. 

32.  16  and  20.  34.   a«6»  and  a*6.         36.   m  and  1. 

rind  the  mean  proportional  to : 

37.  27  and  3.       .  40.  a  +  &  and  a  -  6. 

38.  2  and  18.  41.    a  and  4a. 

39     ^  and  —  ^^*    ^*^^^  and  ll^mn^. 

'     c  h' 

43.  Form  two  proportions  commencing  with  3   from  the 
equation  3x6  =  2x9. 

44.  If  ab  =  Qsy,  form  all  possible  proportions  whose  terms 
are  a,  b,  x,  and  y. 

45.  If  4  a;  =  5  y,  find  the  ratio  oix-.y. 

46.  If  11  a?  =  6  y,  find  the  ratio  of  a; :  y. 

47.  If  ax  =  by,  find  the  ratio  ot  x:y. 

Transform  the  following  equations  so  that  x  will  be  the 
fourth  term : 

48.  3:  5  =  0;: 6.  60.   a;:  2  =  4: 3. 

49.  3:aj  =  5;9.  61.   a:b  =  x:c. 


52. 

5:6  =  ll-x:x, 

53. 

6:rd  =  SS-x:x. 

54. 

6a:  5a  =  22  —  x:x. 

55. 

3:2  =  x:15-x. 

56. 

4:X:5x  =  x:lS  —  x. 

57. 

a:b  =  c  —  x:x. 

58. 

3:4  =  a;_4:4. 

17-4  '       ELEMENTARY  ALGEBRA 

Transform  the  following  proportions  so  that  only  one  term 
contains  x: 

59.  6:5  =  7  +x:x. 

60.  2S:21  =  4:-{-x:x. 

61.  9a:5a  =  20-]-x:x. 

62.  4::7==x:x  +  15. 

63.  3:2  =  4  +  ic:4-a;. 

64.  m:n  =  9  ■i-x:9  —  X. 

65.  a  +  b:a—b=a-\-x:a—x. 

Find  the  ratio  x:y,  ii: 

66.  6x  =  7y.  68.  x:2  =  y:S.       70.    x+y:y  =  7:2. 

67.  y:x  =  7:2.  69.  4::y=S:x.        71.    x  —  y:y  =  2:3. 
72.    a.  +  2/:«^-.7  =  3:l.  ^^^    If  ^  ==  | ,  find  ^±1. 

4^-3y^3  ,g,   If  ^^^^g^^^+^^ 

2a;4-52/     8  2/7'  a;-?/ 

77.  State  whether  the  quantities  mentioned  below  are 
directly  or  inversely  proportional : 

(a)  The  number  of  yards  of  a  certain  kind  of  silk,  and  their 
cost. 

(5)  The  time  a  train  needs  to  travel  10  miles,  and  the  speed 
of  the  train. 

(c)  The  length  of  a  rectangle  of  constant  width,  and  the  area 
of  the  rectangle. 

(d)  The  sum  of  money  producing  $  60  interest  at  5  %,  and 
the  time  necessary  for  it. 

(e)  The  distance  traveled  by  a  train  moving  at  a  uniform 
rate,  and  the  time. 

78.  A  line  4  inches  long  on  a  certain  map  corresponds  to  42 
miles.     A  line  5  inches  long  corresponds  to  how  many  miles  ? 

79.  The  areas  of  two  circles  are  proportional  to  the  squares 
of  their  radii.     The  radii  of  two  circles  are  to  each  other  as 


RATIO  AND  PBOPORTION  175 

3:5,  and  the  area  of  the  smaller  circle  is  18  square  inches. 
Find  the  area  of  the  larger. 

80.  The  temperature  remaining  the  same,  the  volume  of  a 
gas  is  inversely  proportional  to  the  pressure.  A  body  of  gas 
under  a  pressure  of  15  pounds  per  square  inch  has  a  volume 
of  2  cubic  feet.  What  will  be  the  volume  if  the  pressure  is 
6  pounds  per  square  inch  ? 

81.  Two  rectangles  have  equal  areas,  and  the  ratio  of  their 
longer  sides  is  3  :  19.     Find  the  ratio  of  their  shorter  sides. 

82.  The  distance  one  can  see  from  an  elevation  of  h  feet  is 
very  nearly  the  mean  proportional  between  the  elevation  h  and 
the  diameter  of  the  earth  (8000  miles),  provided  these  lengths 
are  expressed  in  the  same  denomination.  What  is  the  greatest 
distance  a  person  can  see  from  a  ship  30  feet  high  ?  From  the 
Eiffel  Tower  (900  feet  high)  ?  From  Mount  Etna  (10,000  feet 
high)?.  

186.  When  a  problem  requires  the  finding  of  two  numbers 
which  are  to  each  other  as  m  :  n,  it  is  advisable  to  represent  these 
unknown  numbers  by  mx  and  nx. 

Ex.  1.  Divide  108  into  two  parts  which  are  to  each  other 
11:7. 

This  problem  contains  two  statements : 
I.    The  sum  of  two  numbers  is  108. 
II.    The  ratio  of  the  same  numbers  is  11  :  7. 

A.  If  we  use  I  to  express  one  unknown  number  by  the  other,  one 
number  is  x,  the  other  108  —  «,  and  II,  written  in  symbols,  produces 
X  :  108  —  X  =  11 :  7,  which  solved  gives  the  value  of  x. 

B.  It  is  better,  however,  to  use  II  to  express  the  two  numbers. 

Let  11  aj  =  the  first  number, 

then  7  05  =  the  second  number. 

I,  expressed  in  symbols,  gives  11  a;  +  7  x  =  108, 

or  18  X  =  108. 

Therefore  x  =  6. 

Hence  11  x  =  66  is  the  first  number, 

and  7  X  =  42  is  the  second  number. 


176  ELEMENTARY  ALGEBRA 


Ex.  2.  A  line  AB,  4  inches  long, 
is  produced  to  a  point  C,  so  that 
(AC):{BC)  =  7:5.    Yind  AC a.ud  BC. 

Let  AC=7x. 

Then  BC=6x. 

Hence  AB  =  2  x. 

Or  '  2  a;  =  4 

x  =  2. 


Therefore  7  x  =  14  =  ^C. 

EXERCISE   73 

1.  Divide  120  in  the  ratio  of  7  : 8. 

2.  Divide  35  in  the  ratio  of  11 :  3. 

3.  Two  numbers  are  to  each  other  as  9:7,  and  their  differ- 
ence is  14.     Find  the  numbers. 

4.  A  straight  line  16  inches  long  is  divided  in  the  ratio  3 : 5. 
What  are  the  parts  ? 

5.  Brass  is  an  alloy  consisting  of  two  parts  of  copper  and  1 
part  of  zinc.  How  many  ounces  of  copper  and  of  zinc  are 
there  in  8  ounces  of  brass  ? 

6.  Gunmetal  consists  of  9  parts  of  copper  and  one  part  of 
tin.  How  many  ounces  of  each  are  there  in  17  ounces  of  gun- 
metal  ? 

7.  Air  is  a  mixture  composed  mainly  of  oxygen  and  nitro- 
gen, whose  volumes  are  to  each  other  as  21 :  79.  How  many 
cubic  feet  of  oxygen  are  there  in  a  room  whose  volume  is  3550 
cubic  feet  ? 

8.  A  line  AB,  9  inches  long,  is  produced  to  a  point  C,  and 
AC  is  to  BC  as  13 :  11.     Find  the  length  of  AC  and  BC. 

9.  The  total  area  of  land  is  to  the  total  area  of  water  as 
7  :  18.  If  the  total  surface  of  the  earth  is  197,000,000  square 
miles,  find  the  number  of  square  miles  of  land  and  of  water. 


BATIO  AND  PROPORTION  177 

10.  Water  consists  of  one  part  of  hydrogen  and  8  parts  of 
oxygen.  How  many  grams  of  hydrogen  are  contained  in  100 
grams  of  water  ? 

11.  Divide  100  in  the  ratio  of  a :  6. 

12.  Divide  a  in  the  ratio  of  6  :  1. 

13.  The  three  sides  of  a  triangle  are  respectively  9,  18,  and 
21  inches,  and  the  longest  side  is  divided  in  the  ratio  of  the 
other  two.     How  long  are  the  parts  ? 

14.  A  line  10  inches  long  is  divided  in  the  ratio  of  a :  2. 
Find  the  parts. 

15.  The  three  sides  of  a  triangle  are  respectively  a,  b,  and  c 
inches.  If  c  is  divided  in  the  ratio  of  the  other  two,  what  are 
its  segments? 

16.  A's  age  is  to  B's  age  as  3  :  2,  and  5  years  ago  the  sum  of 
their  ages  was  45  years.     Find  their  ages. 

17.  A's  age  is  to  B's  age  as  4 : 3,  and  8  years  ago  the  ratio 
of  their  ages  was  3  :  2.     Find  their  ages. 

18.  Two  men,  A  and  B,  start  from  the  same  town  and  travel 
in  the  same  direction.  A's  rate  of  travel  is  to  B's  rate  as  5  :  2. 
Find  their  rates  of  travel,  if  after  three  hours  -they  are  9  miles 
apart. 

19.  One  angle  of  a  triangle  is  67°  and  the  ratio  of  the  two 
others  is  3:6.  How  many  degrees  are  contained  in  these 
angles  ? 

187.  The  products  of  the  corresponding  terms  of  two  or  more 
proportions  are  in  proportion. 

If  ?  =  5  and  ™  =  £,  then«?2  =  5E. 
h     d  n      q  bn      dq 

This  follows  directly  from  §  154. 


178 


ELEMENTARY  ALGEBRA 


Ex.  1.    If  a  :  6  =  3  :  4,  and  6  :  c  =  7  :  5,  find  the  ratio  of  a  :  c. 


a6:&c  =  21:20.     (§187.) 

Or  simplifying, 

a  :  c  =  21  :  20. 

188.  If 

a:6  =  3:4 

and 

6  :  c  =  4  :  5, 

then 

a  :  c  =  3  :  5.     (§  187.) 

These  three  proportions   can  be    conveniently  written   as 


follows : 


a  :  6  :  c  =  3  :  4  :  5. 


Ex.  2.    Divide  100  in  the  ratio  a:b  :c. 


Let 
Tiien 
and 
Hence 
Uniting, 


Therefore 


ax  =  the  first  part. 
bx  =  the  second  part, 
ex  =  the  third  part. 

ax  -{■  bx  +  cx  =  100. 

a;(a  +  &  +  c)=100, 

100 


bx  = 


ex  — 


a^b^-Q. 

100  a 

a-\-b-\-c 

100  6 

a-\-b-^  c 

100  c 

a  +  6  +  c 

Ex.3.    If^  =  5,find   ^^-^y. 
y  3  a;  —  14  2/ 


,  the  first  part. 
,  the  second  part. 
,  the  third  part. 


Dividing  each  term  by  ?/, 
Ix-ly 


VI 


10 


3  aj  -  14  y 


<^" 


^  =5  =  8. 


16  -  U      1 


BATIO  AND  PROPORTION  .  179 

Ex.  4.    If  a  :  6  =  c  :  d,  prove  that  ^-±S^  =  «-!+-2!- 

ab  —  cd     a^  —  & 

According  to  §  169,  this  is  true  if 

a%  +  a'^cd  -  ah(fi  -  cH  =  a^b  +  abc'^  -  a^cd  -cH. 
Or  if  a'^cd  —  ahc^  =  ahc^  —  d^cd.     (Canceling.) 

Or  if  ad— he  =  he  —  ad.     (Dividing  by  ac.) 

Or  if  2  ad!  =  2  he.     (Transposing.) 

Or  if  ad  =  he.     (Dividing  by  2.) 

But  since  the  a(Z  =  &c,     ^l^±^  =  ^t±^. 
ae  —  cd     a'^  —  c^ 

Note.  The  methods  of  simplifying  proportions,  as  given  in  §  184,  will 
frequently  produce  shorter  proofs  than  the  method  used  in  Ex.  4. 

EXERCISE  74 

1.  If  a  :  &  =  3  :  4,  and  6  :  c  =  5  :  6,  find  ct :  c. 

2.  If  a  :  6  =  3  :  a;,  and  6  :  c  =  a; :  5,  find  a  :  c. 

3.  li  P.  Q  =  a:h,  and  Q:  R  =  c:d,  find  the  ratio  P:  R  va. 
terras  of  a,  b,  c,  and  d. 

4.  If  a  :  6  =  2  :  3,  and  &  :  c  =  2  :  9,  find  a  :  c. 

5.  li  a:b  =  m:n,  and  b  :  c  =  m:  n,  find  a  :  c  in  terms  of  m 
and  7*. 

6.  If  a:6  =  2: 1,  6:c  =  2:  3,  and  c  :  d  =  4 :  5,  find  a  :  d. 

7.  If  a  :  5  =  6  :  c  =  1 :  2,  find  a  :  c. 

8.  If  a  :  6  =  6  :  c  =  m  :  71,  find  a  :  c  in  terms  of  m  and  ?i. 

9 .  If  a^ :  i  =  3  : 4,  and  i :  2/'  =  8 :  15,  find  a; :  y. 

2/  '  a;   ^  ^ 

10.  Divide  143  into  three  parts  which  are  to  each  other  as 
2:4:5. 

11.  The  three  angles  of  a  triangle  are  to  each  other  as 
6 :  11 :  19.  Find  the  number  of  degrees  contained  in  each  if 
the  sum  of  the  3  angles  is  180°. 


180  .  ELEMENTARY  ALGEBliA 

12.  A  line  12  inches  long  is  divided  into  3  parts  whicli  are 
to  each  other  as  6  :  7  :  9.     How  long  is  each  part  ? 

13.  A  line  a  inches  long  is  divided  into  3  parts  which  are  to 
each  other  as  3  :  4  :  5.     Find  the  parts. 

14.  The  sum  of  three  sides  (perimeter)  of  a  triangle  is 
90  inches,  and  the  ratio  of  the  3  sides  equals  3:4:5.  Find 
each  side. 

15.  The  perimeter  of  a  triangle  equals  p  inches,  and  the 
ratio  of  the  sides  is  a:  b:  c.     Find  each  side. 

16.  li  x:y=^5:l,  find  2x  +  3yi3gc-2y, 

17.  If  a:6  =  4:3,  find  6a-6:9a-76. 

18.  If  m  :  n  =  5  :  2,  find  8  m  —  7  n  :  4  m  — 13  7i. 

19.  If  a:5--3,  find  6a-26:9a  +  176. 

If  a:  b  =  c:  d,  prove  that : 

20.  ac:bd  =  c^ :  d\ 

21.  a-\-b:  a-\-b-\-c-\-d  =  a:a  +  c. 

22.  a  —  c:b  —  d  =  c:d. 

23.  a'-i-c':b^  +  d^==a^:b'-. 

24.  a^  +  b':c^-^d'  =  d':c^. 

25.  a''h2b^:2b^  =  ac-i-2bd:2bd, 

26.  d'  +  Sb^:a^-3b'  =  c'-{-3d':c^-3d\ 

27.  a^-^b^:ac-{-bd  =  b:d. 

28.  cv^ -\-2b'^:ac-{-2bd=:a:c. 

If  a:b  =  b:  c,  prove  that 

29.  a^:b^  =  ah:bc. 

30.  a'^-^abib^  +  bc^a'ibK 

31.  a  —  b:b  —  c  =  b:c, 

32.  If  a^+b'-+c'+d' :  b^+d'=(f-^d'' :  d^,  prove  that  a :  &=c :  d 


RATIO  AND  PROPORTION  181 

-Solve  the  following  equations : 

33.  8a;-7:6a;4-7  =  12a5-19:9a?4-2. 

34.  5x-24::Sx  +  32  =  7x-^6:9x-S. 

35.  12a!''^-7a?  +  5:18ar^_lla;  +  9  =  2:3. 

36.  8x2_3^_j_4.i2a;2  +  5a;-13  =  2:3. 

37.  X  —  a  :  X —h  =  a -{-b  :  a —  b. 
Hint.     Simplify  before  solving. 

38.  x:  a  —  b  =  x  —  b:  a. 

S9.    a^  —  b^:ax  —  b  =  a-\-b:x. 

40.  x  —  a  +  b:x-\-a  —  b=:a  —  x-\-2b:b  —  x-{-2a. 

41.  Two  travelers  starting  at  two  points  24  miles  distant 
walk  towards  each  other,  and  their  rates  are  as  4:5.  How 
many  miles  does  each  walk  before  they  meet  ? 

42.  The  three  sides  of  a  triangle  are  to  each  other  as  3 :  4  :  5, 
and  the  area  is  24  square  inches.  How  long  is  each  side  ? 
(Compare  §  30.) 

REVIEW   EXEKCI8E   III 

Solve  the  equations : 

1-1  1+  ' 


X  X 


+^ 


_        X      ,      2x  2x   _  3flj*4-4 

x5.     — -  -|- 


x-\-l     2£c-f  3     x-{-2     (x  +  l){2x-\-3){x-^2) 


3. 

a» 

+  b': 

Xes 

■.a'. 

^a6  +  6': 

a  +  b. 

4. 

1 

1 
'2x- 

rs" 

_1. 
4' 

1 
3a;-5 

5.  The  sum  of  two  numbers  is  79,  and  if  the  greater  is 
divided  by  the  smaller,  the  quotient  is  |,  and  the  remainder 
is  2.     Find  the  numbers. 


182  ELEMENTARY  ALGEBRA 

6.  A  boy  bought  some  oranges  at  3^  apiece.  He  sold  ^  of 
them  for  4^  apiece,  and  the  rest  for  5^  apiece,  and  gained  24^. 
How  many  oranges  did  he  buy  ? 

7.  A  man  sold  a  house  for  $4000  more  than  half  its  cost, 
and  gained  thereby  $  1000.     What  did  he  pay  for  the  house  ? 

8.  A  man  sold  a  house  for  $2000  more  than  |  of  the  cost,- 
and  gained  thereby  25  %  on  the  money  invested.  What  did  he 
pay  for  the  house  ? 

m-\-n      . 

^     ci'      vr    '^  — ^  m-\-n 

9.  Simplify \ 

^  m  —  n      .      m  —  n 


m-i-n 

10.  Simplify  f^^ii  -  aM  ^  ^—^,  and  check  the  answer  by 

\_a^  4-1         J     1  +  a^ 

the  numerical  substitution,  a  =  2. 
x-y     x+y 

11.  Simplify  . 


^      x+y      x-y 

x—y     x+y 
-r 


x  +  y      X 


^  x-y  x-\-y 
x-\-y  x-y 
x-y     x  +  y  ^ 


x-\-y     x-y 

12.  Simplify  (5  a^-^+^y*p  -  7  y^"^+^^)  (5  x'^'+Y^  +  7  i/*'"+V). 

13.  Keduce  to  lowest  terms : 

(3  abc  +  A  bed -5  cde)  (36  p^  -  25  q^ 
(6  p  +  5  g)  (6  abc'  - 10  c^de  +  8  bc'd) ' 

14.  Find  two  numbers  whose  sum  equals  a  and  whose  dif- 
ference equals  b. 

15.  Divide  300  into  two  parts  such  that  \  of  the  greater 
increased  by  twice  the  smaller  is  as  much  as  the  smaller 
increased  by  -f  of  the  greater. 

16.  Factor    1.   x*-(a-{-4)x^-\-4:a. 

XL   9{2a-xy-4.(Sa-xy. 

17.  If  a:6  =  3:2,and6:c  =  2:7,finda:c. 


18.    Solve 


RATIO  AND  PROPORTION  183 

c  —  a  b  —  a 


b(c  —  x)      b(b—x) 

19.  Find  the  L.  C.  M.  of: 

(9x'-16yy,9x'-16y%Sx  +  4.y,Sx-4:y, 

20.  If  a  =  5,  find  the  numerical  value  of: 

.7  a'  -  .5  a^  -  .9  a'  +  .7  a^  +  .1  a^  - 1.2  a\ 

21.  Find  the  H.  C.  F.  of: 

a^*"-!,  a^^-l,  (a'^-1)*. 

22.  Find  the  H.C.F.  of: 

23.  Eeduce  to  mixed  or  integral  expressions: 

a'  +  b\ 
a-\-b' 

24.  Simplify  (^""t'^r  ^  ^!^'^  ^  ^H^'. 

25.  Simplify: 

a^(a;  —  5)  (a;  —  c)      b^(x  —  a)  (x  —  c)      c^(x  —  a)(x—  b) 
{a-b)(a-c)  ^  (b  -a){b-  c)  "^    {c-a){c-b)  ' 

26.  Solve: 

3-x     S-x     2-a;^10-a;     x-\-2      5-x 
S  —  x     6  —  x     A  —  x      8  —  x      6  —  x     4  — a; 


2 

X  — 

3y- 

:5. 

y^ 

2x- 
3 

-5 

X: 

=  0, 

y^. 

-|. 

X: 

-i 

y  = 

-*• 

CHAPTER  XI 

SIMULTANEOUS  LINEAR  EQUATIONS 

189.  An  equation  of  the  first  degree  containing  two  or  more 
unknown  numbers  can  be  satisfied  by  any  number  of  values 
of  the  unknown  quantities. 

If                               2a;-32/  =  5.  (1) 

Then  y^ — ^ (^) 

I.e.  if 
If 

If  a;  =  1,  2/  =  —  Ij  etc. 

Hence,  the  equation  is  satisfied  by  an  infinite  number  of  sets 
of  values.     Such  an  equation  is  called  indeterminate. 

However,  if  there  is  given  another  equation,  expressing  a 
different  relation  between  x  and  ?/,  such  as 

«^  +  2/  =  10;  (3) 

these  unknown  numbers  can  be  found. 

From  (3)  it  follows  2/  =  10  —  a;,  and  since  the  equations  have 

to  be  satisfied  by  the  same  values  of  x  and  ?/,  the  two  values  of 

y  must  be  equal. 

Hence  ?^IL£=10-x.  -  (4) 

o 

The  root  of  (4)  is  a;  =  7,  which  substituted  in  (2)  gives  2/  =  3. 
Therefore,  if  both  equations  are  to  be  satisfied  by  the  same 
values  of  x  and  ?/,  there  is  only  one  solution. 

184 


SIMULTANEOUS  LINEAR  EQUATIONS  185 

190.  A  system  of  simultaneous  equations  is  a  group  of  equa- 
tions that  can  be  satisfied  by  the  same  values  of  the  unknown 
numbers. 

jc  +  2  z/  =  5  and  7 x  —  Sy  =  1  ar«  simultaneous  equations,  for  they  are 
satisfied  by  the  values  x  =  l,y  =  2.  But  2x  —  y  =  5  and  4 a:  —  2y  =  6  are 
not  simultaneous,  for  they  cannot  be  satisfied  by  any  value  of  x  and  y. 
The  first  set  of  equations  is  also  called  consistent,  the  last  set  inconsistent. 

191.  Independent  equations  are  equations  representing  differ- 
ent relations  between  the  unknown  quantities  ;  such  equations 
cannot  be  reduced  to  the  same  form. 

6x+  5y  =  50,  and  3 x  +  3 ?/  =  30  can  be  reduced  to  the  same  form  ; 
viz.  x  +  y=  10.  Hence  they  are  not  independent,  for  they  express  the 
same  relation.  Any  set  of  values  satisfying  5  x  +  5  ?/  =  50  will  also  satisfy 
the  equation  3  x  +  3  y  =  30. 

192.  A  system  of  two  simultaneous  equations  containing  two 
unknown  quantities  is  solved  by  combining  them  so  as  to  obtain 
one  equation  containing  only  one  unknown  quantity. 

The  process  of  combining  several  equations  so  as  to  make 
one  unknown  quantit}^  disappear  is  called  elimination. 

193.  The  three  methods  of  elimination  most  frequently  used 
are: 

I.   By  Addition  or  Subtraction. 
II.   By  Substitution, 
III.   By  Gomparison. 

ELIMINATION  BY  ADDITION  OR  SUBTRACTION 

194.  E>.  >.    S.1,.. ="  +  ">-"•  » 


{ 


2x'^Ty  =  -S.  (2) 

Multiply  (1)  by  2,  6  x  -f-  4  y  =  26.  (8) 

Multiply  (2)  by  3,  6  x  -  21  ?/  =  -  24.  (4) 

Subtract  (4 )  from  ( S) ,  26  y  =  60. 

Therefore,  y  =  2. 


186  ELEMENTARY  ALGEBRA 

Substitute  this  value  of  y  in  either  of  the  given  equations,  preferably 
the  simpler  one  (1) ,  3  x  +  4  =  13. 

Therefore  a;  =  3. 

In  general,  eliminate  the  letter  whose  coefficients  have  the  lowest 
common  multiple. 

Check.  3. 3  +  2. 2  =  9  +  4  =  13, 

2.3-7.2  =  6-14  =  -8. 

f   Bx-8y  =  4:7,  (1) 

Ex.    2.      Solve  ^  »  ,  w 

[lSx  +  5y  =  135.  (2) 

Multiply  (1)  by  5,  25  a;  -  15  ?/  =  235.  (3) 

Multiply  (2)  by  3,  39  x  + 15  ?/  =  405.  (4) 

Add  (3)  and  (4),  64x  =  640. 

Therefore  x  =  10.  (5) 

Substitute  (5)  in  (1),  50  -  3  y  =  47. 

Transposing,  —  3  y  =  —  3. 

Therefore  y  =  1. 

jc  =  10. 
Check.     5  .  10  -  3  . 1  =  47, 

13  .  10  +  5  . 1  =  135. 

195.   Hence  to  eliminate  by  addition  or  subtraction: 

Multiply,  if  necessary,  the  equations  by  such  numbers  as  will 
make  the  coefficients  of  one  unknown  quantity  equal. 

If  the  signs  of  these  coefficients  are  like,  subtract  the  equations; 
if  unlike,  add  the  equations. 

EXERCISE  75 

Solve  the  following  systems  of  equations  and  check  the 
answers  : 

'5x+   7^=50,  (12x-lSy  =  9, 


I9x  +  14:y  =  91.  Il7y-   4.x  =  35. 


ri2a; 
*    {l7y- 


SIMULTANEOUS  LINEAR  EQUATIONS 


187 


6. 


7. 


10. 


1. 


12. 


13. 


14. 


15. 


Sx-   5  2/ =  49, 
7  x  +  15y  —  101. 

10a;  +  32/  =  23, 
5y-2x  =  l. 

2x-^5y  =  l, 
6x-\-7y=:3, 

7a;  +  32/=100, 
13  a;—    y=20. 

Sx-15y  =  -S0, 
2x+   Sy  =  W. 

(7x-3y  =  2T, 
[5  x~6y  =  0. 

(7x-Sy=15, 
[5x-\-6y  =  27. 

Sx-4.y  =  ll, 
[5x-3y  =  33, 

'7x-\-    2/  =  42, 
.3x-2y  =  l. 

|3a;  +  42/  =  43,. 
L4a;  +  7?/  =  69. 

(7x-3y  =  23, 
l3a;  +  4y  =  31. 

5x—   7?/  =  —  4, 
[9a;  +  ll2/  =  40. 

x-hy  =  100, 
x-y  =  12. 


16. 


17. 


18. 


19. 


20. 


21. 


22. 


24. 


26. 


r  9a:  +  142/  =  83, 
l39a;-35y  =  -23. 

(2x-lly=:-95, 
1    a;-   32/  =  0. 

r3a;-30  2/  =  15, 
L2a;4-102/  =  40. 

ri3a;-ll2/  =  131, 
ll9a;-24  2/  =  33. 

r8a;  +  32/  =  37, 
[Sy-3x  =  50. 


x  —  5y  =  4, 


3x-\-5y  =  32. 

fl6a;-152/  =  18, 
2x  +  52/  =  16. 

(4:X  —  7y  =  —  5y 
\5x+    y  =  f 

|Ja;+i.y  =  6, 
l3a;-42/  =  4. 

(1.5x-2y==ly 
{2.5x-3y  =  6. 


26.  (i^-2^  =  ^^ 

r4.9y-3.2a;  =  j 

27.  J 
\3.5  2/-2.4a;  =  . 

1.5  a; -1.1 2/ =  .01, 


1.9, 
9. 


( i.ox  —  i.iy  = 
[    2x-1.7y  = 


,08. 


188 


ELEMENTARY  ALGEBRA 


ELIMINATION  BY   SUBSTITUTION 

196.    Solve  (2x-7y  =  -8, 

.3a;  +  22/  =  13. 

Transposing  —  1  y  in  (1)  and  dividing  by  2,         x  =  — ^ 
Substituting  this  value  in  (2),       ^1    '  ~  ^]  +  2  ?/ 
Clearing  of  fractions, 


(1) 
(2) 


13. 


21  ?/  -  24  +  4  ?/  =  26. 
26  2/  =  50. 
Therefore  y  =  2. 

This  value  substituted  in  either  (1)  or  (2)  gives  x  =  S, 

197.    Hence  to  eliminate  by  substitution : 

Find  in  one  equation  the  value  of  an  unknown  quantity  in 
terms  of  the  other.  Substitute  this  value  for  one  unknown  quan- 
tity in  the  other  equation,  and  solve  the  resulting  equation, 

EXERCISE  76 
Solve  by  substitution : 
f  3  a;  +  2  2/  =  5, 
[2x  +  hy  =  7. 
4.x-   7y=     2, 
12x^25y  =  ---2. 
j9y-5x='2, 
l32/-h4x  =  29. 
21a5-92/  =  -3, 
Sx  +  By=   31. 
'20?/-   9a;=   26, 
.  40  2/  -  27  a;  =  -  2. 
(S5x-17y=   59, 
I    7x-   5y  =  ~9. 


2. 


a. 


4. 


5. 


6. 


8. 


9. 


10. 


11, 


12. 


13. 


7. 


r6y-   7x=     35, 
I  24  V  -  41  a;  =  -  29. 


14. 


r5a;  +  5    =10?/, 
I5x-2y  =  16. 
2x=   5y-ll, 
Sx  =  27-7y. 
4t/  =  3a;  +  4, 
5y  =  4:X  +  3. 
lla;=    Sy-h   7, 
lla^  =  13?/-23. 
|aj  +  52/  =  29, 
U  +  3?/=:19. 

Uaj  +  J^  =  12. 
ia;  +  iy  =  18, 
ia;-i2/  =  10. 


SIMULTANEOUS  LINEAR  EQUATIONS 


189 


ELIMINATION   BY   COMPARISON 


198.   Ex. 

From  (1), 

From  (2), 

Therefore 

Clearing  of  fractions, 
Transposing  and  uniting, 
Therefore 
Substitute  in  (4), 


Solve  ^        ' 

I       x-7  =  -12y. 


a;  =  7  -  12  y. 

2  +  2y 

3 
2  +  2  y  =  21 

38  y-  19. 


a) 

(2) 
(3) 
(4) 


=  7  -V2y. 
36  y. 


199.   Hence  to  eliminate  by  comparison : 

In  each  equation  find  the  value  of  one  unknown  quantity  in 
terms  of  the  other.  Form  an  equation  with  these  values  and 
solve  it. 


EXERCISE  77 


Solve  by  comparison : 

9y  =  2x-Sl, 
9y  =  5  —  16x. 


2. 


r    x  +  4.y  =  37, 
\2x-\-5y  =  53. 


4. 


6. 


7a;  +  32/  =  100, 
3a;-    y  =  20. 

'5a;  +  3?/  +  2  =  0, 
.3x  +  22/  +  l=0. 

+  66  =  0, 
+  13  =  0. 


8. 


9. 


ri0a;4-7y  +  4  =  0, 
1  6x  +  5y-\-2  =  0. 

nx  +  iy=:6, 
l3a;-4?,=4. 

(5x-4..9y  =  \, 
l3a;-2.9'?/  =  l. 

111 


y 

7x-10y  =  .l, 
162/ =  .1. 


r21a;+    Sy 
1 23?/ -28  a; 


Ex.     Solve 


190  ELEMENTABY  ALGEBRA 

200.  Whenever  one  unknown  quantity  can  be  removed 
without  clearing  of  fractions,  it  is  advantageous  to  do  so ;  in 
most  cases,  however,  the  equation  must  be  cleared  of  fractions 
and  simplified  before  elimination  is  possible. 

^  +  ^-±3.3,  (1) 

^-^  =  1.  (2) 

Multiplying  (1)  by  12  and  (2)  by  14, 

4a;  +  8  +  32/  +  9  =  36.  (3) 

7  X  +  21  -  2  2/  -  4  =  14.  (4) 

From  (3),  4  x  +  3  y  =  19.  (5) 

From  (4),  7x-2y=-  3.  (6) 

Multiplying  (5)  by  2  and  (6)  by  3, 

Sx  +  6y  =  SS.  (7) 

21x-6y=-9.  (8) 

Adding  (7)  and  (8),  29  x  =  29. 

x  =  l. 
Substituting  in  (6),  7  -2y  =  -S. 

y  =  6. 
Check.    l+J     5_+3^.      2  =  3 
3  4 

LL§_§-±-^  =  2-i  =  1 

2  7 

EXERCISE  78 

Solve  the  following  systems  of  equations  by  any  method, 
and  check  the  answers  of  Exs.  1-10: 


r2(a;  +  3)+3(2/  +  4)  =  26, 
U(a;4-5)+5(iy4-6)  =  64. 
r6(a:-7)-7(2/-8)  =  18, 
l8(a;-9)-9(2/-10)  =  26. 
r2(3a;-4)-f-3(42/-5)  =  43, 
(4(5  a; -  6) -h 5(6 2/ -  7)  =  121. 


SIMULTANEOUS  LINEAR  EQUATIONS 


191 


10. 


2.3  x  + 4.7  2/ =  70, 

3.4  a; +  5.6  2/ =  90. 


\3(x- 


2)  -  3(1/ +  1)  =  23, 
(x-2)  +  5{y-l)=:19. 


6. 


(S{2x-y)-^4.(x-2y)  =  ST, 
\2(3x-y)-3(x-y)  =  S2. 


x-^2y     2x-j-y 
7  5 


3a;-2 
1 


2 


3a;  +  l     52/  +  4 

1^2 
4a;_3     7?/-6' 


7a;-13 
[Sy-5 


4. 


'15a;  +  l^g 


45  —  2/ 

12^+19^25 
a;~10 


11. 


12. 


13. 


14. 


15. 


3a;  +  1^4 
4-22/     3' 
x  +  y  =  l. 

l-2xJ6 
5-32/     2' 
2/  —  a;  =  4. 

a;-3^2 

y  +  2     3' 
«-fjL^3 
2/-2      2' 

'a;  +  2y  +  l_ 


2x-y-\-\ 
3a;  — t/  +  l 


2, 


=  5. 


a;-2^  +  3 

^±3l±13_^3 
4a;  +  52/-25       ' 
8a;  +  2/  +  6    ^g 
5a;  +  32/-23 


16. 


a;  +  l     y  +  2^2(a;-y) 
3  4  5 


a?-3     y  — 3 


=  22/~aj. 


17. 


3£--2j/     5£-3j/^       . 
5^3  ' 

?-|32/  +  4_£^.^^^^ 


192 


ELEMENTARY  ALGEBRA 


18. 


19. 


20. 


21. 


22 


(.T-4)(.v-f7)  =  (:»-3)Cv  +  4), 
(a^-f5)(^-2)  =  (.T  +  2)(2/-l). 

(a^  +  3)(2/  +  5)  =  (a;4-l)(y  +  8), 
(2a^-3)(5^  +  7)  =  2(5x-6)(^  +  l). 

I  15^12-^' 
]7^_5j^3_ 
[  25      16      20* 

'3a;H-4y     5y  — 3a;_2a;  +  3y     3y  — 2g 
12  9        ~        6     '         '   3       ' 

I  1  /     ,   .V   ,    1  /       ox  '  3  X     V  — 21 

-(cc  +  4)-+ (w  — 3)  = "^— 

[4^         ^10^       ^       4  2 


X  :  y  =  3  :  4, 


23. 


(a'  +  4):(2/  +  l)=2:3, 


(a^-l):(2/  +  2)  =  l:2.        ""       l(a;  + 2)  :  (2/- 1)  =  3  : 1. 
24.    (x  +  l):(?/  +  l):(.'C  +  ?/)  =  3:4:5. 

201.  In  many  equations  it  is  advantageous  at  first  not  to 
considi^r  x  aiid  y  as  unknown  quantities,  but  some  expressions 

involving  x,  and  y,  such  as  -  and  -,  xy,  etc. 

X  y 


(1) 

(2) 
(8) 


Ex.  1.     Solve  • 

X      y 
15     4_ 

a;      2/~ 

=  3. 

:4 

2x(2), 

0)  +  (3), 

X 

Clearing  of  fractions, 

83  =  llx. 

3C=:3. 

SIMULTANEOUS  LINEAR  EQUATIONS 


19a 


Substituting  x  =  3*  in  (1), 


y 


y 

Therefore  ?/  =  4. 

Check,     f +  1  =  1  +  2  =  3;  Jj5._|  =  5_l  ^4. 

Examples  of  this  type,  however,  can  also  be  solved  by  the  regular 
method,  provided  they  do  not  involve  more  than  two  unknown  quantities. 

Clearing  (1)  and  (2)  of  fractions,  3  ^  +  g  x  =  3  a:y.  (4) 

15y  — 4x  =  4xy.  (5) 

2x(5),                                  SOy-Sx  =  Sxy.  (6) 

(4) +  (6),                                           33y  =  nxy.  (7) 
Dividing  by  11  y,                                   3  =  x,  etc. 

1 


=  2. 


6an/  +  | 


Ex.  2.     Solve 

6x(l), 

(2) -(3), 

Therefore 

Substituting  x  =  1  in  (1), 


21. 


6 


6  x«/  -  -  =  12. 

X 


?  =  9. 

X 

x  =  l. 

y  -  1  =  2. 

2/ =  3. 


(1) 

(2) 
(3) 


Solve 


1. 


2. 


a;     2/ 

5 
^6' 

1_ 

1_ 

1 

a? 

y 

'6 

12 

X 

+1 

=  5, 

16 
a; 

2/ 
3 
0 

=  2. 

EXERCISE  79 


■^  -f  -  ae  —  O, 

a;      2/ 


11 

X 


7^3 
i     2* 


6.   i 


6. 


a;     y 


2_5 
a;     2/ 


-7. 


5  +  5  =  28, 
X     y 


6     4' 


0. 


194 


ELEMENTARY  ALGEBRA 


7.   i 


9. 


r7_l2 

X      y 

-  =  ?- 

[y      X 


^' 


X       y 

48     43     ^ 

— = h2JL. 


fl2 

a; 
10 
a; 


10. 


11. 


12. 


3a;--  =  6. 

9a;--  =  0, 

2/ 

6a;  +  ?  =  ll. 

y 


16 


ra^2/  +  7a;  =  ll, 
1 5  a;?/ +  4  ic  =  24. 


17. 


13. 


14. 


15. 


^i-4-i  =  2 
3a;     4?/       ' 

a;     2?/ 
X     2y 


17_5 
6a;     2/ 


'    -^=12. 


a;-l     2/-2 
4   .+_^=22. 


a;-l     2/-2 


a;?/     y 
a;?/     2/ 


11 
3' 


LITERAL   SIMULTANEOUS  EQUATIONS 


?02.   Ex.  1.     Solve 

a)xn, 
(2)  X  6, 
(3) -(4), 
Uniting, 

Dividing, 

(l)xm, 
(2)xa, 
(7) -(6), 
Uniting, 


mx-{- 


by  =  c. 
ny=p. 

anx  +  buy  =  en. 
bmx  +  bny  =  bp. 
anx  —  bmx  =  en  —  bp. 
(an  —  bni)x  =  en  —  bp. 
^  _  en  —  bp 


an  —  bm 
amx  +  bmy  =  cm. 
amx  +  any  =  ap. 
any  —  bmy  =  ap  —  em. 
(an  —  bm)y  =  ap  —  cm. 

ap  —  cm 


(1) 
(2) 
(3) 
(4) 


(6) 

(6) 
(7) 


y  = 


an  —  bm 


SIMULTANEOUS  LINEAR  EQUATIONS 


195 


EXERCISE    80 


Solve 
1, 


4. 


5. 


1.7. 


18. 


ax  -\-2by  =  6, 
2  ax -\- 5  by  =  ^. 
ax  -\-by  =  a, 
bx  —  ay  =  b. 
ax  —  by  =  0, 
ex  —  dy  =  1. 
x-\-y  =  a, 
x  —  y  =  b. 
ax-\-y  =  my 
x  —  y  =  n. 

X  +  my  =  a, 
x~ny  =6. 
ax-\-by  =  c, 
dx  +  ey=j: 

ax-\-by  =  c, 
mx  =  ny. 

(a-\-b)x—  {a 

(a  -\-b)x-{-  (a 

(  (a-\-  c)x  —  (a 

L  (a-\-b)y—  (a 


9. 


10. 


11. 


12. 


13. 


14. 


a  (a;  4- 2/)  =  16, 
b{x-y)=^. 
ax-\-by  =  1, 
bx  +  ay  =  0. 
ax  +  by  =  4:, 
ax—by  =  2. 

a     b 

7  +  ^  =  2- 
c     a 

a;  4-1 
=  a, 

y  +  1 

X 

x_a 
y     b 

x  +  1 


=  b. 


15. 
16. 


((a 


-  4-  -  =  0, 
a;     2/ 

1      b 
=  a. 

«    y 

a;      2/ 
X     y 


b)y 
■h)y 

c)2/ 
b)x. 

19. 
20. 


2/4-1 
=  4a&, 
=  2(a^  +  b^ 
=  2a6. 


0  +  1 

&4-1' 


2ac. 

a  .  6 
-4--  =  c, 
a;     y 

m  ,  n 
X      y     ^ 

ax  —  by  =  ay 
ab{x  —  l)r=y. 


196  ELEMENTARY  ALGEBRA 

21.  Solve  the  equations 

ax  -j-by  =  Cf 
px-\-qy  =  r, 

and  from  the  results  find  by  numerical  substitution  the  roots  of 
Exs.  1-4,  page  188. 

22.  Find  a  and  s  in  terms  of  n,  d,  and  I  if 

[l  =  a  +  (n~l)d. 

23.  From  the  same  simultaneous  equations  find  d  in  terms 
of  a,  n,  and  I. 

24.  From  the  same  equations  find  s  in  terms  of  a,  d,  and  I. 

25.  From  the  same  system  of  equations  find  n  in  terms  of 
a,  I,  and  s. 

SIMULTANEOUS    EQUATIONS    INVOLVING   MORE    THAN 
TWO   UNKNOWN    QUANTITIES 

203.  To  solve  equations  containing  three  unknown  quantities 
three  simultaneous  independent  equations  must  be  given. 

*  By  eliminating  one  unknown  quantity  from  any  pair  of  equa- 
tions, and  the  same  unknown  quantity  from  another  pair,  the 
problem  is  reduced  to  the  solution  of  two  simultaneous  equations 
containing  two  unknown  quantities. 

Similarly  four  equations,  containing  four  unknown  quanti- 
ties, are  reduced  to  three  equations  containing  three  unknown 
quantities,  etc. 

Ex.  1.    Solve  the  following  system  of  equations: 

2  a;  -  3  2/  +  4  2;  =  8.  (1) 

3aj4-4  2/-52  =  -4.  (2) 

4  a;  -  6  2/  +  3  2  =  1.  (3) 


SIMULTANEOUS  LINEAR  EQUATIONS 


197 


Eliminate  y. 

Multiplying  (1)  by  4, 

Sx-12y  +  16z=     .32 

Multiplying  (2)  by  3, 

9  X  +  12  2/  -  15  ^  =  -  12 

Adding, 

17  a;             +      z=     20 

(4) 

Multiplying  (2)  by  3, 

9x  +  12y-iryz=-12 

Multiplying  (3)  by  2, 

Sx  -V2y  -}-    ()z=        2 

Adding, 

17x              -    9;s=-10 

(6) 

Eliminating  x  from  (4) 

and  (5). 

(4) -(5), 

10^  =  30. 

Therefore 

0  =  3. 

(6) 

Substitute  this  value  in 

(4),           17  a;  +  3  =  20. 

Therefore 

a:  =  l. 

(7) 

Substituting  the  values  of  x  and  z  in  (1), 

2  -  3  y  +  12  =  8. 

3?/  =  6. 

Hence 

2/  =  2. 

(8) 

Check.     2.1-3.2  +  4.3  =  8;   3.1  +  4.2-5.3=--4; 

4.1-6.2  +  3.3  =  1. 

204.  After  some  practice  the  student  will  be  able  to  elimi- 
nate the  unknown  quantities  of  simple  examples  mentally. 
Eliminating  by  subtraction  should  then  be  avoided,  and  by  the 
use  of  negative  multipliers  every  elimination  should  be  based 
upon  addition. 


Ex.  2.    Solve   (1)  2  a;  +  3 1/  +  2  ;2  =  21 

(2)  5x-2j/  +  30  =  16 

(3)  1x-6y  +  4z  =  lb 


3 
-2 


(4)  -4x+13?/  =  31 

(5)  -    x+    7?/ =  19 


Multipliers    for    the 
elimination  of  z. 


Multipliers  for  the  elirai- 
nation  of  x. 


-  15y  =  -45. 

Therefore 

2/  =  3. 

From  (5), 

a  =  2. 

From  (1), 

«  =  4. 

198 


ELEMENTARY  ALGEBRA 


EXERCISE   81 


Solve; 

(x  +  5y  +  6z  =  29, 

1.  I  10 a? 4- 2/ +  2:2  =  18,  10. 
[5x-i-9y-{-3z  =  32. 

2x-\-5y-3z=^13, 

2.  6x-3y  +  4:Z=:16,  11. 
,5x-{-3y-6z  =  15. 
(2x-{-3y-z  =  21, 

3.  ]  9x-^5y  —  2z  =  71,  12. 
[6x  —  7y-\-5z  =  55. 
(x-\-2y-\-3z  =  32, 

4.  ]  2  a; +  2/ +  3  2  =  31,  13. 
[4.x~2y  +  z  =  12. 

2x-3y-^4.z  =  S, 

5.  3x  +  4:y  —  5z  =  —  4:,  14. 
Ax-6y-^3z  =  l. 
(7x-6y  +  2z  =  4:, 

6.  I  14  05  — 8  2/ -2  =  0,  15. 
[2x-9y-3z  =  -35. 

6x-{-5y-4.z  =  9S,- 

15y  +  6z  =  -6,         16. 
Sx-3y-9z==12. 
9x-7y-6z  =  18j 
12x-Uy-i-9z  =  27,         17. 
18a;-352/  +  152:  =  0. 
3x-y-i-z  =  7, 
x  +  2y  —  4:Z  =  —  8,  18. 

2x-2y  +  z  =  2, 


(6x 
7.    '  9a; 


8. 


9. 


x-\-y-i-z  =  12, 
4:X-\-3y  +  5z  =  4:9y 
5x  —  2y-i-z  =  l. 
x-\-y-]-z  =  9, 
x  —  y-\-z  =  —  l, 
x-\-y  —  z  =  —  5. 
12x-5y-\-lSz^30y 
2x  —  5y-\-Sz  =  15, 
4:X  +  y  -\-4:Z  =  3. 
5x-\-y  —  4:Z  =  0j 
3x-{-2y-{-3z  =  110, 
2x-3y  +  z  =  0. 
2x  +  4.y-\-Sz  =  13, 
x  +  y-{-z  =  3, 
3x  +  9y-\-27z  =  34:. 
x-\-2y~{-3z  =  32j 
2x-\-3y-\-z  =  4:2, 
3x-\-y-\-2z  =  A0. 
3x  +  3y-\-z  =  17, 
3x-^y  +  3z  =  15, 

[x  +  3y-{-3z  =  13. 

'2x  —  3y  —  4:Z  =  6, 
3x-4.y-{-5z  =  56, 

[5x  +  7y-9z  =  106. 
x-{-2y-i-3z  =  10, 
3x-5y-9z  =  -10, 

I2x-3y=:0. 


SIMULTANEOUS  LINEAR  EQUATIONS 


199 


19. 


20. 


21. 


22. 


23. 


29. 


5  ?/ —  7  «  4- 4  2;  =  ■ 

-44, 

z-5y  =  -S3, 

5y-x  =  22. 

x  +  y=28, 

y  +  z  =  25, 

X  +  Z  =  24:. 

3a;  +  42/  =  25, 

52/4-62  =  50, 

7z  +  Sx  =  59. 

x-{-y  +  z=:3% 

13a;  =  102/, 

16y  =  13z. 

2x  +  3y-4.z  = 

-3, 

y  =  5x-7, 

z  =  7x-10. 

24. 


25. 


26. 


27. 


3x-4:y  +  5z  =  lS, 

X  =  4:Z—17f 

y  =  oz  —  21. 

^  +  ^  +  ^  =  24, 

3  5     4 

4  3     5       ' 

^-^  +  ^  =  6. 
6     15     10 

lAx-2Ay-3z  =  -A. 
.8a;+3.2  2/  +  42  =  11.2, 
x-2y-\-l.Sz=7.6. 

.lxi-.2y-{-.3z  =  3.2, 
.2x-\-.ly-{-.3z  =  3.1, 
.3x  +  .2y  +  .lz  =  S. 


28. 


5(2a;-32/)+3(3a;-52)  =  31, 
2(2  a;-  3  2/>+  5 (3  ?/  -  7  ;2)=  -  45, 
I  4(3  a;- 5  2/) -3(5  2/- 7 2)  =  55. 


=  10, 
=  9, 


x  +  y 
y  —  z 

x_-\-_z 
x-y 

xi-5 


31. 


(x+3:y  +  z=:2:l, 

30.   \y-{-3:x  +  z  =  l:ly 

[x  +  3:x  +  y  =  l:2. 


3y-2x  .  4z-3x,2y-7z_f. 
-^— +  -^— +  — g— -0, 


5x 


-3y     4:y-3x     7y-\-2z 
15  5  9 


'T7> 


7g     6.T  — 52/.3z 


8 


12 


=  -5f}. 


200 


ELEMENTARY  ALGEBRA 


32. 


33. 


34. 


35. 


36. 


37. 


38. 


X     y      z 

5-4  =  3, 

X     y 


^z      y 


51 


-  +  -  +  i  =  12, 

X     y      z 

i  5-1  +  5  =  18, 

X     y      z 


5_3 

I  a;     y 


=  13. 


ri , i_  5 
I    ~  ^' 

X     y      b 


1_1 

2      12 


5x  —  y  +  Sz=af 
5y  —  z-{-3x  =  b, 
5z—x-}-3y  =  c. 

(7  x  +  lly-\-z  =  a, 
7y-{-llz-\-x  =  af 
7z-\' 11  x-\-y  =  c, 

x-\-y-\-z  =  56, 
x:y  :z=l:2:  5. 

x-\-y  =  2aj 
y-^z  =  2c, 

\  z+x  =  2b. 


39. 


( ax  +  by  =  2  ah, 
by-{-cz  =  ah  -\-  <?, 
y^-\-y-\-z  =  (i-{-h-\-c. 


fl.l       9. 

-  +  -  =  2  a, 
2/      ^ 

40. 

1  +  ^  =  26, 
X      z 

1  +  1  =  20. 

'  ax -^hy  -\-  cz=  a -\-hj 

41.   \x-\-dy  =  d, 

{x^z^l. 

42.    ^ 


'3it'  +  6?/  +  22;  +  v  =  2, 
a?  —  2/  —  32;  —  4'y  =  3, 

x-^2y-2z-2v  =  <), 
y2x-^y  —  z—3v  =  h. 


43. 


2.T  +  ^-22;  =  0, 
3£c  +  42/  +  3v  =  l, 
I4v  — 5a;  +  32;4-?/  =  lc 


44. 


a^  +  y 


5, 


xy 


=  7. 


yz 


46. 


«="  •  a^ 


x  +  z  =  6. 


SIMULTANEOUS  LINEAR  EQUATIONS  201 

PROBLEMS  LEADING  TO   SIMULTANEOUS  EQUATIONS 

205.  Problems  involving  several  unknown  quantities  must 
contain,  either  directly  or  implied,  as  many  verbal  statements 
as  there  are  unknown  quantities.  If  the  problem  is  to  be 
solved  by  one  equation  containing  only  one  unknown  quantity, 
whose  symbol  is  x,  all  statements  except  one  are  used  to  ex- 
press the  unknown  quantities  in  terms  of  x,  while  the  remain- 
ing statement  produces  the  equation.     (§  IOlI.) 

In  complex  examples,  however,  it  is  often  very  difficult  to 
express  some  of  the  unknown  quantities  in  terms  of  one  of 
them.  In  such  a  case  it  is  advisable  to  represent  every 
unknown  quantity  by  a  different  letter,  and  to  express  every 
verbal  statement  as  an  equation. 

Ex.  1.  The  sum  of  the  three  digits  of  a  number  is  8.  The 
digit  in  the  tens'  place  is  J  of  the  sura  of  the  other  two  digits, 
and  if  396  be  added  to  the  number,  the  first  and  the  last  digit 
will  be  interchanged.     Find  the  number. 

Obviously  it  is  very  difficult  to  express  two  of  the  required  digits  in 
terms  of  the  other ;  hence  we  employ  3  letters  for  the  three  unknown 
quantities. 

Let  X  =  the  digit  in  the  hundreds'  place, 

y  =  the  digit  in  the  tens'  place, 
and  z  =  the  digit  in  the  units'  place. 

Then      lOOx  +  lOy -{-  z  =  the  number. 

The  three  statements  of  the  problem  can  now  be  readily  expressed  in 

«y°^^«^«^  x  +  y  +  z  =  S.  (1) 

y  =  U^  +  ^)'  (2) 

100  X  +  10  y  +  0  +  396  =  100  5:  +  10  y  +  x.  (3) 

The  solution  of  these  equations  gives  x  =  l,  y  =  2,  z  =  5. 
Hence  the  required  number  is  125. 

Check.        1  +  2  +  5  =  8;  2  =  ^1+5);  125  +  396  =  521. 

206.  Between  the  two  methods  of  expressing  all  unknown 
quantities  in  terms  of  one  letter,  x,  and  employing  as  many 


202  ELEMENTARY  ALGEBRA 

letters  as  there  are  unknown  quantities,  there  is  often  the 
intermediate  way  of  representing  some,  but  not  all,  unknown 
quantities  in  different  letters.  E.g.  denote  two  quantities 
directly  by  letters,  as  x  and  y,  and  express  the  remaining 
unknown  quantities  in  terras  of  x  and  y  by  some  of  the  given 
verbal  statements.  The  remaining  two  statements  give  the  two 
necessary  equations. 

E.g.  in  the  example  of  the  preceding. paragraph,  we  could 
make  x  =  the  first  digit,  and  y  =  the  last  digit.  Then,  according 
to  the  second  statement,  the  second  digit  =  ^(x-\-  y),  and  the 
first  and  third  verbal  statements,  expressed  in  symbols,  give 

x  +  ^(x  +  y)-^y  =  S. 

100a;  +  -i^(a;4-2/)+^  +  396  =  1002/  +  -y(«^  +  2/)  +  «^- 
These  equations  lead  of  course  to  the  same  number  as  before. 

Although  this  method  is  often  somewhat  shorter  than  the 
method  previously  described,  it  is  advantageous  only  if  the 
example  is  simple.  In  most  complex  examples,  the  equations 
can  be  written  with  the  least  difficulty  if  each  unknown 
quantity  is  represented  by  a  different  letter. 

Ex.  2.  If  both  numerator  and  denominator  of  a  fraction  be 
increased  by  one,  the  fraction  is  reduced  to  |;  and  if  both 
numerator  and  denominator  of  the  reciprocal  of  the  frac- 
tion be  diminished  by  one,  the  fraction  is  reduced  to  2.  Find 
the  fraction. 

Let  X  =  the  numerator, 

and  ,  y  =  the  denominator. 

then   -  =  the  fraction.     By  expressing  the  two  statements  in  symbols, 

y 

we  obtain,  x+l_2  ,.. 

and  y-^  =  2.  (2) 

X—  1 


These  equations  give  x  =  S  and  y  =  5.     Hence  the  fraction  is  f . 

3  +  1^4^2.    5-1^4 
5  +  1     6     3'    3-  1     2 


Check.     ^±^  =  ^  =  ^:   ^l  =  i  =  2. 


SIMULTANEOUS  LINEAR  EQUATIONS 


203 


Ex.  3.  A,  B,  and  C  travel  from  the  same  place  in  the  same 
direction.  B  starts  two  hours  after  A  and  travels  one  mile  per 
hour  faster  than  A.  C,  who  travels  2  miles  an  hour  faster 
than  B,  starts  2  hours  after  B  and  overtakes  A  at  the  same 
instant  as  B.     How  many  miles  has  A  then  traveled  ? 


Time 

(Hours) 

Eatb 

(Miles  per  hour) 

Distance 
(Miles) 

A 

X 

y 

xy 

B 

x-2 

y  +  i 

xy  +     x  —  2y  -  2 

C 

X-4: 

2/  +  3 

xy-{-Sx-4y~12 

. 

Since  the  three  men  traveled  the  same  distance, 
xy  =  xy  +  X  —  2  y  —  2. 
xy  =  xy  -]-  Sx  —  iy  —  12. 

Or  x-2y  =  2. 

Sx-4y  =  V2. 

(4)  -  2  X  (3)  a;  =  8. 

From  (3)  y  =  S. 

Hence  xy  =  24  miles,  the  distance  traveled  by  A. 

Check.     8x3  =  24,  6  x  4  =  24,  .4  x  6  =  24. 


(1) 
(2) 
(3) 
(4) 


EXERCISE  82 

1.  Three  times  a  certain  number  increased  by  five  times 
another  number  equals  31,  and  the  second  one  increased  by  one 
is  equal  to  three  times  the  first  one.     Find  the  numbers. 

2.  Find  two  numbers  whose  sum  and  whose  quotient  equal  5. 

3.  Half  the  sum  of  two  numbers  is  11,  and  the  fifth  part  of 
their  difference  is  2.     Find  the  numbers. 


204  ELEMENTARY  ALGEBRA 

4.  If  1  be  added  to  the  numerator  of  a  fraction,  its  value 
is  -I-  If  1  be  added  to  its  denominator,  the  fraction  is  reduced 
to  \.     Find  the  fraction.     (See  Ex.  2,  §  206.) 

5.  If  3  be  added  to  both  terms  of  a  fraction,  its  value  is  ^. 
If  1  be  subtracted  from  both  terms,  its  value  is  i  Find  the 
fraction. 

6.  If  the  numerator  of  a  fraction  is  trebled,  and  its  denomi- 
nator diminished  by  2,  it  is  reduced  to  |.  If  the  denominator 
is  doubled,  and  the  numerator  increased  by  3,  the  fraction  is 
reduced  to  i.     Find  the  fraction. 

7.  A  fraction  is  reduced  to  i,  if  both  its  terms  are  increased 
by  1,  and  8  times  the  numerator  diminished  by  3  times  the 
denominator  equals  3.     What  is  the  fraction  ? 

8.  Find  two  fractions,  with  numerators  2  and  3  respect- 
ively, whose  sum  is  ||,  and  such  that  when  their  denominators 
are  interchanged,  their  sum  is  1^. 

9.  The  sum  of  the  digits  of  a  number  of  two  figures  is  10, 
and  if  3  times  the  units'  digit  is  added  to  the  number,  the 
digits  will  be  interchanged.  What  is  the  number  ?  (See  Ex. 
1,  §  205.) 

10.  The  sum  of  a  number  of  two  digits  and  of  the  number 
formed  by  reversing  the  digits  is  121,  and  five  times  the  tens' 
digit  exceeds  the  units'  digit  by  one.     Find  the  number. 

11.  The  sum  of  the  three  digits  of  a  number  is  11,  and  the 
sum  of  the  first  two  digits  exceeds  the  last  digit  by  1.  If  27 
is  added  to  the  number,  the  last  two  digits  are  interchanged. 
Find  the  number. 

12.  A  number  consists  of  three  digits,  the  sum  of  the  last 
two  digits  is  two  less  than  five  times  the  first  digit ;  three  times 
the  second  digit  exceeds  the  first  digit  by  one ;  and  if  the  first 
di-git  be  subtracted  from  the  number,  the  remainder  is  215. 
Find  the  number. 


SIMULTANEOUS    LINEAR    EQUATIONS  205 

13.  If  a  certain  number  be  divided  by  the  snm  of  its  three 
digits,  the  quotient  is  43.  The  sum  of  the  first  two  digits  is 
equal  to  the  third  digit,  and  if  99  be  added  to  the  number, 
the  first  and  last  digits  will  be  interchanged.  Find  the 
number. 

14.  If  a  certain  number  be  divided  by  the  sum  of  its  two 
digits,  the  quotient  is  4  and  the  remainder  3.  Three  times  the 
first  digit  exceeds  the  second  digit  by  3.     Find  the  number. 

15.  Twice  A's  age  exceeds  the  sum  of  B's  and  C's  ages  by 
30,  and  B's  age  is  ^  the  sum  of  A's  and  C's  ages.  Ten  years 
ago  the  sum  of  their  ages  was  60.     Find  their  present  ages. 

16.  Five  years  ago  A  was  as  old  as  B  will  be  in  10  years ; 
and  10  years  ago  B  was  as  old  as  C  will  be  in  5  years.  If  the 
sum  of  their  ages  is  60,  how  old  now  is  each  ? 

17.  A  sum  of  $10,000  is  partly  invested  at  5%,  partly  at 
4  %,  and  partly  at  3  %,  bringing  a  total  yearly  interest  of  $390. 
The  5  %  investment  brings  $  10  more  interest  than  the  4  %  and 
3  %  investments  together.  How  much  money  is  invested  at 
3  %,  4  %,  and  5  %  respectively  ? 

18.  A  sum  of  money  at  simple  interest  amounted  in  2  years 
to  $  330,  in  5  years  to  $  375.  What  was  the  sum  and  the  rate 
of  interest  ? 

19.  A  sum  of  money  at  simple  interest  amounted  in  6  years 
to  $  16,000,  and  in  8  years  to  $  17,000.  What  was  the  sum  of 
money  and  the  interest  ? 

20.  A  sum  of  money  at  6%  simple  interest  amounted  in  a 
certain  time  to  $  944.  In  half  the  time  at  4%  it  would  amount 
to  $  348.     Find  the  sum  and  the  time.  * 

21.  A  sum  of  money  at  simple  interest  amounted  in  a  years 
to  m  dollars,  and  in  b  years  to  ?i  dollars.  What  was  the  sum  ? 
Find  the  sum  if  a  =  3,  6  =  5,  m  =  $  500,  and  n  =  $  600. 


206  ELEMENTARY  ALGEBBA 

22.  The  sums  of  $  1200  and  $  1400  are  invested  at  different 
rates  and  their  annual  interest  is  $  111.  If  the  rates  of  inter- 
est were  exchanged,  the  annual  interest  would  be  f  110.  Find 
the  rates  of  interest. 

23.  Three  sums  of  money  are  respectively  invested  at  4%, 
4%,  and  5%,  and  their  annual  interest  is  $1280.  If  the  first 
and  third  were  invested  at  4%,  and  the  second  at  5%,  the 
interest  of  the  second  would  be  $  350  less  than  the  interest 
of  the  other  two.  If  the  rates  of  interest  were  respectively 
5%,  4%,  and  3|-%,  the  first  sum  would  bring  $380  less  inter- 
est than  the  other  two.     What  are  the  sums  ? 

24.  If  a  rectangle  has  the  same  area  as  another  3  feet  longer 
and  2  feet  narrower,  and  the  same  area  as  a  third  rectangle 
which  is  8  feet  longer  and  4  feet  narrower,  what  are  its  dimen- 
sions ? 

25.  If  a  rectangle  were  100  feet  longer  and  25  feet  narrower^ 
its  area  would  contain  2500  square  feet  more.  If  it  were  100 
feet  shorter  and  50  feet  wider,  its  area  would  contain  5000 
square  feet  less.     What  are  the  dimensions  of  the  rectangle? 

26.  If  a  rectangle  were  1  foot  longer  and  1  foot  narrower,  its 
area  would  be  m  square  feet  less.  If  it  were  1  foot  longer 
and  one  foot  wider,  its  area  would  be  n  feet  more.  What  are 
the  dimensions  of  the  rectangle  ? 

27.  A,  B,  and  C  working  together  can  do  a  piece  of  work  in 

1  day ;  A  and  C  together  in  li  days,  and  B  and  C  together  in 

2  days.    In  how  many  days  can  each  do  the  same  work  alone  ? 

28.  A  and  B  together  do  a  piece  of  work  in  18  days ;  A  and. 
C  in  16  days,  and  B  and  C  in  20  days.    In  how  many  days  can 

each  alone  do  the  same  work  ? 

• 

29.  A  and  B  together  can  do  a  piece  of  work  in  a  days ;  B 
and  C  in  6  days,  and  C  and  A  in  c  days.  In  how  many  days 
can  each  alone  do  the  same  work  ?  Find  the  answer  if  a  =  2, 
&  =  3,  c  =  4. 


SIMULTANEOUS  LINEAR  EQUATIONS  207 

30.  A  farmer  sold  a  number-  of  horses,  cows,  and  sheep,  for 
f  500,  receiving  $100  for  each  horse,  $50  for  each  cow,  and 
$15  for  each  sheep.  The  number  of  sheep  was  twice  the 
number  of  horses  and  cows  together.  How  many, did  he  sell 
of  each  if  the  total  number  of  animals  was  fifteen  ? 

31.  The  sum  of  the  3  angles  of  any  triangle  is  180°.  If 
one  angle  of  a  triangle  exceeds  half  the  sum  of  the  other  two 
angles  by  15°  and  half  their  difference  by  65°,  what  are  the 
angles  ? 

32.  The  difference  of  two  angles  of  a  triangle  is  equal  to  the 
third  angle,  and  their  sum  is  ^^-  of  the  third  angle.  What  are 
the  angles  ? 

33.  A  and  B  received  together  $  107  wages  for  working  25 
and  16  days  respectively.  If  A  had  worked  24  days  and  B 
had  worked  20  days,  they  would  have  received  $112.  What 
were  the  daily  wages  of  each  ? 

34.  The  perimeter  (i.e.  the  sum  of  the  sides)  of  a  triangle 
is  39  inches.  The  greatest  side  is  7  inches  less  than  the 
sum  of  the  other  two,  and  one  of  these  two  is  twice  as  large 
as  the  difference  of  the  remaining  two.  Find  the  length  of 
each  side. 

35.  On  the  three  sides  of  a  triangle  ABO,  respectively,  three 
points  D,  E,  and  F,  are  taken  so 
that  AD  =  AF,  BD  =  BE,  and  CE 
=  CF.  UAB=Q  inches,  BO  =4: 
inches,  and  AC=  8  inches,  what  is 
the  length  of  AD,  BE,  and  CF? 

Note.  If  a  circle  is  inscribed  in  the 
triangle  ABC  touching  the  sides  in  Z>,  E, 
and  F  (see  diagram),  then  AD  =  AF, 
BD  =  BE,  and  CE=CF. 

36.  A  circle  is  inscribed  in  triangle  ABC  touching  the  three 
sides  in  D,  E,  and  F.  Find  the  parts  of  the  sides  if  AB  —  5, 
50=7,  and  (7.4  =  8. 


208  ELEMENTARY  ALGEBRA 

37.  A  circle  is  inscribed  in  a  triangle  ABC  (see  diagram  of 
Ex.  35)  whose  perimeter  is  14  feet,  AD  exceeds  FC  by  2  inches, 
and  BD  equals  1  inch.     Find  the  sides  of  the  triangle. 

38.  On  the  sides  of  a  triangle  ABC 
the  points  D,  E,  and  F  are  so  taken  that 
AD  is  4  times  as  large  as  AF,  BE  is  3 
times  as  large  as  BD,  and  CF  is  5  times 
as  large  as  CE.  If  AB  =  5  inches,  BC 
=  4  inches,  and  CA  =  6  inches,  find  the 
parts  of  the  three  sides. 

39.  In  the  annexed  diagram  angle  a  =  angle  b,  angle  c  = 
angle  d,  and  angle  e  =  angle  /.  If  angle  ABC  =60°,  angle 
BAC  =  40°,  and  angle  BCA  =  80°, 
find  angles  a,  c,  and  e. 

Note.     0  is  the  center  of  the  circum- 
scribed circle. 

40.  The  sum  of  the  radii  of  two 
circles  is  15  inches,  and  the  differ- 
ence of  their  circumferences  is  44. 
If  TT  is  taken  equal  to  3|,  what  are 
the  radii  ?  (The  circumference  of 
a  circle  0,  whose  radius  is  E,  is  determined  by  the  formula 
C=2itR.) 

41.  The  sum  of  the  radii  of  two  circles  is  r  inches,  and  the 
difference  of  their  circumferences  is  d  inches.     Find  the  radii. 

42.  Two  persons  start  to  walk  in  the  same  direction  from 
twd  stations  12  miles  apart,  and  one  overtakes  the  other  after 
6  hours.  If  they  had  walked  toward  each  other,  they  would 
have  met  in  2  hours.     What  are  their  rates  of  travel  ? 

43.  Two  persons  start  to  walk  in  the  same  direction  from 
two  stations  d  miles  apart,  and  one  overtakes  the  other  after  a 
hours.  If  they  had  walked  toward  each  other,  they  would  have 
met  in  h  hours.     What  are  their  rates  of  travel  ? 


SIMULTANEOUS  LINEAR  EQUATIONS  209 

44.  A  takes  2  hours  longer  than  B  to  travel  12  miles,  but  if 
A  would  double  his  pace,  he  would  walk  it  in  one  hour  less  than 
B.     Find  their  rates  of  walking. 

45.  A  takes  2  hours  longer  than  B  to  travel  d  miles,  but  if 
A  should  double  his  pace,  he  would  walk  it  in  1  hour  less. 
Find  the  time  B  needs  to  walk  the  distance. 


INTERPRETATION  OF  NEGATIVE   RESULTS  AND  THE 

0     a 

O'    O'    00 


FORMS  OF   ^     "^      ^ 


207.  The  results  of  problems  and  other  examples  appear 
sometimes  in  forms  which  require  a  special  interpretation,  as 
a     0      0_ 


0'   0'   ~'  **"• 


208.  Interpretation  of  -.      According,  to  the   definition  of 

division,  -  =  x,ii  0  =  0  x.    But  this  equation  is  satisfied  by  any 

finite  value  of  x,  hence  -  may  be  any  finite  number,  or  -  is 
indeterminate. 

209.  Interpretation  of  ^.     The  fraction  -  increases  if  x  de- 

u  X 

creases ;  e.g.  -^  =  100  a,    — ^ —  =  10,000  a.      By  making   x 

T077        ^  T^TTTRT 

sufficiently   small,    -   can  be  made  larger  than  any  assigned 

X 

number,  however  great.     If  x  approaches  the  value  zero,  -  be- 

X 

comes  infinitely  large.    It  is  customary  to<represent  this  result 
by  the  equation  ^  =00. 

Note.      The    symbol    oo    is    called    infinity.      In    some    examples, 

the  result   -  is  only  indeterminate  in  form  ;  e.g.  It  x  =  3,  ^  —  4  a;  -f  o 

Q  0  a:^  —  5  a;  +  6 

would  be  -,  if  we  substitute  directly.     By  reducing  the  fraction  to  its 

lowest  terms,  \^  ~  ^K^  ~  v  =^  ~  ■^    and  then  substituting,  we  obtain 
for  2.  '  (x-'S)ix-2)      x-2' 

p 


210  ELEMENTARY  ALGEBRA 

210.  Interpretation  of  — .     The  fraction  -  decreases  if  x  in- 

OO  X 

creases,  and  becomes  infinitely  small,  if  x  is  infinitely  large. 
This  resnlt  is  usually  written  : 

?-  =  Q. 

GO 

211.  The  discussion  of  a  problem  determines  the  nature  of 
the  solutions,  if  the  given  quantities  assume  all  possible  values. 
The  following  example  illustrates  the  discussion  of  a  problem : 

Two  couriers,  A  and  B,  travel  by  the  same  road  in  the  same 
direction,  and  at  12  o'clock  B  is  d  miles  in  advance  of  A.  If 
A  travels  a  miles  per  hour,  and  B  travels  b  miles  per  hour, 
after  how  many  hours  will  A  overtake  B  ? 

Suppose  they  meet  after  x  hours,  then  A  has  traveled  ax, 
and  B  bx,  miles.  But  since  A  has  traveled  d  miles  more  than  B, 
ax  —  bx  =  d, 

therefore,  x  = 

a—  b 

Discussion.  1.  If  a  is  greater  than  b,  the  value  of  x  is  posi- 
tive, and  A  will  overtake  B  after  12  o'clock. 

2.  If  a  is  smaller  than  b,  the  value  of  x  is  negative,  e.g.  If 
a  —  2,  6  =  3,  d  =  4,  then  a;  =  —  4,  i.e.  the  men  do  not  meet 
after  12  o'clock,  but  they  were  together  4  hours  before  12 
o'clock.  This  is  obvious  from  the  data  of  the  problem,  for  if 
A  walks  more  slowly  than  B,  he  cannot  overtake  B. 

Hence  there  is  no  answer  to  the  problem  stated  above.  To 
make  a  solution  possible  the  problem  should  read.  How  many 
hours  before  12  o'cloick  did  they  meet  ? 

3.  If  a  =:  b,  then  x=  -  =  oo ;  i.e.  A  and  B  will  never  meet, 

and  evidently  two  men  traveling  at  the  same  rate,  and  d  miles 
apart,  will  never  meet.     Hence  the  problem  has  no  solution. 

4.  li  a  =  b,  d  =  0,  then  x  =  -  =  any  finite  number,  as  2,  3,  4, 

etc.,  i.e.  A  and  B  are  always  together.     This  also  is  obvious 
from  the  nature  of  the  problem. 


SIMULTANEOUS  LINEAR  EQUATIONS  211 

212.  Negative  solutions  frequently  indicate  a  fault  in  the 
enunciation  of  the  problem. 

213.  The  result  -  or  oo  indicates  that  the  problem  has  no  solu- 
tion. If  in  an  equation  all  terms  containing  the  unknown  quan- 
tity cancel  J  while  the  remaining  terms  do  not  cancel,  the  root  is 
infinity. 

214.  The  solution  a;  =  -  indicates  that  the  problem  is  indeter- 
minate, or  that  X  may  equal  any  finite  number.  If  all  terms  of  an 
equation^  ivithout  exception,  cancel,  the  answer  is  indeterminate. 


EXERCISE  83 

Interpret  the  answers  of  the  following  problems;  and  if 
negative  solutions  occur,  indicate  what  changes  in  the  state- 
ment of  the  problem  would  make  a  solution  possible. 

1.  A  is  25  years  old,  and  B  is  15  years  old.  How  many 
years  hence  will  A  be  twice  as  old  as  B? 

2.  Four  times  a  certain  number  increased  by  12  equals  four 
times  the  excess  of  the  number  over  2.     Find  the  number. 

3.  One  half  of  a  certain  number  exceeds  the  sum  of  its  third 
and  sixth  part  by  12.     Find  the  number. 

4.  Find  3  consecutive  numbers  such  that  the  square  of  the 
second  exceeds  the  product  of  the  first  and  third  by  1. 

5.  One  half  of  a  certain  number  is  equal  to  the  sum  of  its 
fourth,  sixth,  and  twelfth  part.     Find  the  number. 

6.  a  times  a  certain  number  increased  by  b  equals  c  times 
the  number  increased  by  d.  Find  the  general  answer  and  inter- 
pret the  answer,  if 

(a)  a  =  c,  b  and  d  are  unequal. 

(b)  b  =  d,  a  and  c  are  unequal. 

(c)  a  =  c,  and  b  =  d. 


CHAPTER   XII 
INVOLUTION 

215.  Involution  is  the  operation  of  raising  a  quantity  to  a 
positive  integral  power. 

To  find  (3  a^Y  is  a  problem  of  involution.  Since  a  power 
is  a  special  kind  of  product,  involution  may  be  effected  by 
repeated  multiplication. 

216.  Law  of  Signs.     According  to  §  50. 

-\-a'  +  a'  -\-a  =  -\-o?. 

—  a-  —a'  =^-{-a^. 

—  a-  —a  —  a  —  —  a^  etc. 
Obviously  it  follows  that 

1.  All  powers  of  a  positive  quantity  are  positive. 

2.  All  even  poivers  of  a  negative  quantity  are  positive. 

3.  All  odd  powers  of  a  negative  quantity  are  negative. 

(—  a)6  is  positive,  (—  ah'^y  is  negative. 

INVOLUTION  OF  MONOMIALS 

217.  According  to  §  52. 

1.  {a^Y:=a'^'a'^'a^  =  or+^+^  =  a^. 

2.  (65)4  ^  55  .  56  .  55  .  ^,5  ^  55+5+6+5  _  520. 

3.  (a'*)"*  =  a»  .  a"  •••  to  TO  factors 

^^1^  ^n+w+n+n  to  m  terms 

=  a*"". 

4.  ( -  3  a%^)  4  =  ( -  3  a%^)  •  ( -  3  ^253) .  ( _  3  ^258)  .  (_  3  ^258) 

--=  81  «86i2. 


6     /     2?n2\3^      (2m2)3^      8m6 
*    V      3  w5  j  (3  n^y         27  n}^' 


212 


INVOLUTION  213 

To  find  the  exponent  of  the  power  of  a  powers  multiply  the  given 
exponents. 

To  raise  a  product  to  a  given  power ^  raise  each  of  its  factors  to 
the  required  power. 

To  raise  a  fraction  to  a  power,  raise  its  terms  to  the  required 
power. 

EXERCISE  84 

Perform  the  operations  indicated : 
ay,  20.    (~2a"*6")«. 


1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 
10. 
11. 
12. 
13. 
14. 
15. 
16. 
17. 
18. 
19. 


33. 


f     7mV 


\x^y-z 


-o:T-  21.  {~Sa%h^f. 

^yy-  22.  (ay.  34. 

-^2/^.  23.  (-ay-.                        /    -1  V 

'^'^^'  25.  f^Y.  36.    (^)\ 

3mny.  \3j  \a^) 

-5pV)^  2e.  f-2^Y.  37.  ^^niY. 

^2pqy.  V     3  6;  VSa-W 

-4ayc)3  27.  (-?^J.  38.    (- A.J. 

3a6W)'.  28.  (^^r^j-  39.   (-g^J- 

29.  (^s)  •  40. 

-|a6W.  V    ^      /  V       ^V 


(-¥)'■ 


214  ELEMENTARY  ALGEBRA 

INVOLUTION   OF  BINOMIALS 

218.  The  square  of  a  binomial  was  discussed  in  §  65. 

219.  The  cube  of  a  binomial  we  obtain  by  multiplying  (a  -f-  6)^ 
hja  +  h.  {a  +  bf  =  a'  +  Za''b  +  ^ab''+b\ 

and  (a  -bY  =  a^-Z  a'b  +  3  ab'  -  b\ 

Ex.  1.    Find  the  cube  of  2x  +  ^y. 

(2  X  +  3  2/)3  =  (2  x)3  +  3(2  a:)2(8  y)  +  3(2  a:)  (3  2/)2  +  (3  y)3 
=  8  x3  +  36  x^y  +  54  xy"-  +  27  y'^. 

Ex.  2.    Find  the  cube  of  3  x^-y"". 

(3  X2  -  ?/n)3  ^  (3  ^.2)3  _  3(3  a;2)2(^n)  +  3(3  x2)(2/'^)2  _  (yn)3 

=  27  x6  -  27  x4«/«  +  9  a;22/2«  -  j/S". 

EXERCISE    85 
Perform  the  operations  indicated  : 

1.  (m-hn)3.  9.    (3  a +  1)3.  17.  (4a;2-5/)^ 

2.  (.r-2/f.  10.    (5a2  +  l)3.  18.  (3a3-2  6c)l 

3.  (a; -7)3.  11.    ila^-lf.  19.  (5ajy-3^2)3. 

4.  {x-^yf.  12.    (5a;+2?/)3.  20.  (4  0^7/2  - 1)3. 

5.  (l  +  2a;)2.  13.    (3a;-5  2/)l  21.  {2x-yy. 

6.  (l+2a;)3.  14.    (3a^/-l)3.  22.  {x^^-y^^. 

7.  (l-3a^)2.  15.    {ax-{-hy)\  23.  (2a'' -3)3. 

8.  (l-3a;=^)3.  16.    {2ax-Zhyy.  24.  (2  a*"  -  3  6'')3. 

220.   The  higher  powers  of  binomials,  frequently  called  expan- 
sions, are  obtained  by  multiplication,  as  follows : 

{a  +  hy  =  a'-\-2ab  +  h\ 

(a  +  by  =  a3  +  3  a'b  +  3  aft^  +  b\ 

(a  +  6)*  =  a*  +  4  a^b  +  6  a^b''  +  4  a63  +  b\ 

(a  -\.by  =  a'  +  5  a'b  + 10  a^b'  + 10  a'b''  -i-5ab*  +  b',  etc. 


INVOLUTION  215 

An  examination  of  these  results  shows  that : 

1.  The  number  of  terms  is  one  greater  than  the  exponent  of  the 
binomial. 

2.  The  exponent  of  a  in  the  first  term  is  the  same  as  the  expo- 
nent of  the  binomial,  a7id  decreases  in  each  succeeding  term  by  one. 

3.  The  exponent  of  b  is  1  in  the  second  term  of  the  result ,  and 
increases  by  1  in  each  succeeding  term. 

4.  The  coefficient  of  the  first  term  is  1. 

5.  The  coefficient  of  the  second  term  equals  the  exponent  of  the 
binomial. 

6.  The  coefficient  of  any  term  of  the  power  multiplied  by  the 
exponent  of  a,  and  the  result  divided  by  1  plus  the  exponent  of  6, 
is  the  coefficient  of  the  next  term. 

Ex.1.   Expand  (x  4- ?/)^ 

=  x^  +  bx^y  + 10  x^y^  +  10  x^y^  -\- h  xy^ -\- j^. 

Ex.  2.   Expand  (x  —  yf. 
{x-yY  =  x^^  5x*(-  y)  +  10x3(-  yY+\Ox\-  y)8+  5a;(-  yY 

=  x^  -5x^y  +  10  x^y^  -  10  x^y^  -\-5xy^-y^. 

221.  The  signs  of  the  last  answer  are  alternately  plus  and 
minus,  since  the  even  powers  of  —y  are  positive,  and  the  odd 
powers  negative. 

Ex.  3.   Expand  (2x^-Sf)*. 
(2  3^  -  3fy  =  {2  a^'  -  4.(2  x'f(Sf)  +  6(2  a^\S  fy 

=  16  a^  -  96  a;y  +  216  a;V  -  216  a;^^,^  +  81 2/^1 


216 


ELEMENTARY  ALGEBRA 


Expand : 

EXERCISE  86 

1.    {a-h)\ 

14, 

[m  4-  nf. 

27. 

(mn  +  ly. 

2.    (m  +  1)*. 

15. 

[m  —  rif. 

28. 

(a-\-bcy. 

3.    {1-ny. 

16. 

[a  -  x)\ 

29. 

(a^-\-b'cy. 

4.    {ah  +  c:)\ 

17. 

[m  —  7i)\ 

30. 

(2a-iy. 

5.    (^-2/^y. 

18. 

iX  +  ay. 

31. 

(2a-iy. 

6.  (i  +  dy. 

19. 

[a-iy 

32. 

(i  +  2xy. 

7.    (m  —  w)^ 

20. 

[1-ay, 

33. 

(Sa^  +  S/. 

8.    {a  +  xf. 

21.    < 

[I  -  ay. 

34. 

(3  a'  -  by. 

9.    (a;-&)^ 

22.    { 

[a  - 1)8. 

35. 

(2a^-5fy. 

10.    (h  +  x)\ 

23. 

[a-2y. 

36. 

(2a  +  5cy. 

11.    (a  +  6)«. 

24. 

[1  -  m'y. 

37. 

(^+5fy. 

12.    (a;-7i)«. 

25. 

[1-m'y, 

38. 

(3x  +  2yy. 

13.    (x  +  yy. 

26.    ( 

v+iy. 

39. 

(2x^-5  fy. 

INVOLUTION  OF  POLYNOMIALS 

222.  The  square  of  polynomials  was  discussed  in  §  67. 

223.  The  higher  powers  of  polynomials  are  found  either 
by  multiplication,  or  by  transforming  the  polynomials  into 
binomials. 

Ex.  1.   Expand  (a +&  — c)^ 
(a  +  6-c)3=[(a  +  6)-c]3 

=  (a  +  6)8-3(a  +  6)2c+3(a+6)c2-c8 

=a3+3  a26+3  afcH&^-S  c(a2+2  a6  +  &-2)+3  ac^^Zhd^-c^ 

=  a3+3  a264-3  ah'^+h^-ZcC^c-Q  abc-S  b^c-\-S  ac^+S  hc^-cK 


INVOLUTION  217 

Ex.2.    Expand  (ar^- 3  ar^-2x-l)^ 

(a;3_3a;2-2x-l)3 
:[(a;3-3x2)-(2x  +  l)]3 

:  (x3  -  3  x2)3  _  3(x3  -  3^2)2(2 X  +  1)  +  3(x3  _  3  x2)(2 X  +  1)2  -  (2  X  +  1)3 
:x9-9x8+27x7-27x6-3(x6-6xH9x*)(2x+l)+3(x3-3x2)(4x2+4x+l) 

-(8x3  + 12x2 +  6x+l) 
:  x9  -  9  x8  +  27  x7  -  27  x6  -  6  x7  +  33x6  -  36  x5  -  27  x*  +  12  x^  -  24  x* 

-  33  x3  -  9x2  -  8  x3  -  12  x2  -  6  X  -  1 
:  a:9  _  9  a;8  +  21  x^  +  6  x6  -  24  x5  -  51  X*  -  41  x3  -  21  x2  -  6  X  -  1. 

EXERCISE  87 
Expand : 

1.  (a-f-6  +  c)2.  10.  (a^-x-iy. 

2.  (a  +  &-c-d)2.  11.  (a  +  b+c  +  df. 

3.  (x-2/-3cH-5d)2.  12.  (a-6  +  c-d)3. 
4.'  (m2-2w2-3^2_,_4^)2^  13  (<^  _  5  _  ^  _  c?)3. 

5.  (a^-3aj2  +  4a;-l)2.  14.    (ar^  +  a;^  +  x  + 1)". 

6.  (a  +  &  +  c)^  15.    {a'+a  +  iy. 

8.  (a;  +  2/-2;f. 

9.  (aj2  +  a;  +  l/. 


(-'-^j 


CHAPTER  XIII 
EVOLUTION 

224.  Evolution  is  the  operation  of  finding  a  root  of  a  quantity ; 
it  is  the  inverse  of  involution. 

Va  =  X  means  x"  =  a. 

\/27  =  y  means  y^  =  27,  or  y  =zS. 

y/b^  =  X  means  x^  =  b^^,  or  x  =  6*. 

225.  It  follows  from  the  law  of  signs  in  involution  that  i 

1.  Any  even  root  of  a  positive  quantity  may  he  either  positive 
or  negative. 

2.  Every  odd  root  of  a  quayitity  has  the  same  sign  as  the 
quantity, 

\/9  =  +  3,  or  -3  (usually  written  ±3);  for  (  +  3)2  and  (-3)2  equal  9. 

sT^  =  -  3,  for  (-  3)3  =  _  27. 

\/a*  =  ±  a,  for  (+  ay  =  aS  and  (-  a)*  =  a*. 

v^  =  2,  etc. 

226.  Since  even  powers  can  never  be  negative,  it  is  evidently- 
impossible  to  express  an  even  root  of  a  negative  quantity  by 
the  usual  system  of  numbers.  Such  roots  are  called  imaginary 
numbers,  and  all  other  numbers  are,  for  distinction,  called  real 
numbers. 

Thus  V—  1  is  an  imaginary  number,  which  can  be  simplified  no 
further. 

218 


EVOLUTION  219 

EVOLUTION   OF  MONOMIALS 

227.   The  following  examples  are  solved  by  the  definition  of 
a  root : 

Ex.  1.    y/a^^  =  ±  a^  for  (±  a*)*  =  a^^. 

Ex.  2.    \/a^  =  a"*,  for  (a**)"  =  a"*". 


Ex.  3.    V8  a6ft%i2  =  2  a^dc*,  for  (2  a^bc^^  =  8  a^b^c^^. 


Ex.4.   ^/81^!!^  =  ±^,  for  /±3aBfty^81ai^ 
Ex.5.    {IZI^^^-I^,  for   f-l«!-V  = 


32  gis 

243  6iOc2o" 


228.  To  extract  the  root  of  a  power ^  divide  the  exponent  by  the 
index. 

A  root  of  a  product  equals  the  product  of  the  roots  of  the  factors. 

To  extract  a  root  of  a  fraction,  extract  the  roots  of  the  numerator 
and  denominator. 

EXERCISE  88 

Simplify  the  following  expressions : 
1.    ^/r.  12.    a/27  a%^c\ 


21.     ^, 

2.    V25.  13.    ^^^27.  ^^^"''^ 


Va*.  14.  V- 27  a?bK  22.  ^/IH^". 

3 ^  a^ 

^'    ^^'  15.  a/ -512  a V.  23.  -^-Sfa^. 

5.    Va^.  '        ;^g^  VlUx^.  24  i"/^^"- 


®-    ^^'-  17.    -^16"^^ 

7.    ■v/7«.  10. 25.    V^iV^, 

15.    v^  a  .  n  being  even. 


8.  VTOO. 

9.  ^-1000.  19-    V-Te"* 


26.    ^2^ 

n  being  odd. 


10.    V4.a^b\ 


11.    V9a;y. 


20.    x1i^^  27.    V/?^^^. 


220  ELEMENTARY  ALGEBRA 


28.  V(-a)'.  32.    V(a  +  6H-c/.  36.  ^/ia-^-bf^ 

o, „  , X  being  even 

29.  {^-a)\  33.    VaV(x  +  2/)'.  / 

37.  V-^2. 

30.  V(M^^.  34.    ^-8(a.-2/)«.  ^^  ^^7^^. 

31.  Va'4-2a6  4-6l   35.    -v/-32(m+?iy^.  39.  V.49pl 


40.  V.027a^-  42.    -\/27  a=^  -  V  -  8  a^  -  Val 

41.  VlGo'H  ^v^^4a"^  43.    V4~a¥« - ^/So^P. 

EVOLUTION  OF  POLYNOMIALS   AND  ARITHMETICAL 
NUMBERS 

229.  A  trinomial  is  a  perfect  square  if  one  of  its  terms  is  equal 
to  twice  the  product  of  the  square  roots  of  the  two  other  terms. 
(§  118.)  In  such  a  case  the  square  root  can  be  found  by  in- 
spection. 

Ex.  1.    Find  the  square  root  of  a?^  —  6  ^y^  +  9  ?/*. 

aj6  _  6  ^f  +  9  2/^  =  (ar^  -  3  f)\     (§  118.) 

Hence       Va;«  -  6  a^2/^  +  9  ?/*  =  ±{x^-Z  /). 

EXERCISE   89 

Extract  the  square  roots  of  the  following  expressions : 

1.  l-4a4-4a2.  7.  ^a^-^^aV ^-Vllh\ 

2.  a*  +  16  62-8a26.  8.  16  ic^  -  8  a^y  +  2/'. 

3.  o>-\-l-2a^.  9.  o}'^h^-2(jC-h\ 

4.  16«^-l-aH8a^  10.  36  a^- 120  a^^c-f  225  6V. 

5.  1+49  2/^-14/.  11.  49  m«- 140  771^7124- 100  71^ 

6.  y?y^-^xyz-\-'^z^.  12.  81  a^y - 126 a:^^^^^ _^ 49  2«. 

13.  a2  +  62^c2  +  2a6  +  2ftc  +  2ac. 

14.  a2  4-62  +  c2  +  2a&-26c-2ac. 

15.  a2  +  62^1-2a-2a&  +  26.    . 


EVOLUTION  221 

230.  In  order  to  find  a  general  method  for  extracting  the 
square  root  of  a  polynomial,  let  us  consider  the  relation  of  a-|--6 
to  its  square,  a^  -\-2  ah-\- 1/. 

The  first  term  a  of  the  root  is  the  square  root  of  the  first 
term  a^. 

The  second  term  of  the  root  can  be  obtained  by  dividing  the 
second  term  2  ah  by  the  double  of  a,  the  so-called  trial  divisor ; 

2  ah      , 

a  +  6  is  the  root  if  the  given  expression  is  a  perfect  square. 
In  most  cases,  however,  it  is  not  known  whether  the  given 
expression  is  a  perfect  square,  and  we  have  then  to  consider 
that  2  ah  -[-W  =  h{2a-\-h)j  i.e.  the  sum  of  trial  divisor  2  a,  and 
h,  multiplied  by  h  must  give  the  last  two  terms  of  the  square. 

The  work  may  be  arranged  as  follows : 


a2  +  2a6  +  62  |  gj^h 


2a-f  6 


2ah-\-W 
2ah  +  h^ 


Ex.  1.   Extract  the  square  root  of  16  «^  -  24  cc^^^  +  9  f. 


16  x*  -  24  icV  +  9  1/6  I  4  a;''  -  3  y8 
16  a;* 


8  x2  -  3y3 


-  24  a:2?/3  +  9 ; 

-  24  a;2y3  +  9 


Explanation.  Arrange  the  expression  according  to  descending  powers 
of  X.  The  square  root  of  16  x*  is  4  x"^.,  the  first  term  of  the  root. 
Subtracting  the  square  of  Ax^  from  the  trinomial  gives  the  remainder 
—  24  x'^y^  +  9  ?/6.  By  doubling  4  a;^,  we  obtain  8  a;^,  the  trial  divisor. 
Dividing  the  first  term  of  the  remainder,  —  24  x^y^.,  by  the  trial  divisor 
8  x2,  we  obtain  the  next  term  of  the  root  —  3y8,  which  has  to  be  added  to 
the  trial  divisor.  Multiply  the  complete  divisor  8  a;^  —  3  y^  by  —  3  y^,  and 
subtract  the  product  from  the  remainder.  As  there  is  no  remainder, 
4  a;2  —  3  2/8  is  the  required  square  root. 


222 


ELEMENTARY  ALGEBRA 


231.  The  process  of  the  preceding  article  can  be  extended  to 
polynomials  of  more  than  three  terms.  We  find  the  first  two 
terms  of  the  root  by  the  method  used  in  Ex.  1,  and  consider 
their  sum  one  term,  the  first  term  of  the  answer.  Hence  the 
double  of  this  term  is  the  new  trial  divisor;  by  division  we 
find  the  next  term  of  the  root,  and  so  forth. 


Ex.  2.   Extractthesquarerootof  16a^-24aH4-12a+25a2. 
Arranging  according  to  descending  powers  of  a. 

16  a*  -  24  a3  +  25  a2  -  12  a  +  4 
16  a* 


4  a2  -  3  a  -f  2 


Square  of  4  a^. 
First  remainder. 
First  trial  divisor,  8  a^. 

First  complete  divisor,  Sa^  —  Sa.       

Second  remainder. 

Second  trial  divisor,  S  a^  —  6  a. 

Second  complete  divisor,  S  a^  —  6  a  -^  2. 


24  a3  +  25  a2 
24  a3  +  9  ^2 


12a +4 


+  16  a2  _  12  «  +  4 
+  16  a2  _  12  a  +  4 


As  there  is  no  remainder,  the  required  root  is  ±  (4  a^  —  3  a  +  2). 

232.    When  some  terms  of  a  polynomial  are  fractional,  the 
one  which  has  the  greater  power  in  the  denominator  is  con- 

sidered  the  lower  power,  thus  3  x^  —  5  -{-  —  —  2  x  -\-  -  arranged 
according  to  descending  powers  of  x  equals,  ^ 


3x'-2x 


X     ar 


Ex.   3.   Extractthesquarerootof  4a^H-25-  — -i2a;4-— . 

X  Qi? 

Arranging  according  to  descending  powers  of  x. 


4a; 


4a; 


4a;2_i2a;  +  25-?^  +  l^ 
4a;2                           ^       ^' 

2a;- 

3  +  i 

X 

-3 

-12  a; +  25 
-12  a; +  9 

%  — 

6  +  ^ 

X 

+  16-24  +  18 
X       x^ 

EVOLUTION  223 

EXERCISE  90 

Extract  the  square  roots  of  the  following  expressions: 

1.  9r*-24r2s-f-16s2. 

2.  a*-hl3a^-12a-6a^  +  4:. 

3.  Sa^  +  a'-2a-2a^  +  l. 

4.  9a!*-12ar^  +  34a;2_20a;  +  25. 

5.  49  a' -A2a%  + 37 a'b' -12 ab^  +  4:b\ 

6.  2ab-2ac-2bc-^a'-{-b'-i-(^. 

7.  9a2  +  24a3  +  46a*  +  40a^  +  25a«. 

8.  9a2-24a3-14a*  +  40a^  +  25a«. 

10.  25a2  +  30a«  +  69a^  +  36a*-f  36a«. 

11.  49a2  +  42a6  +  962_56ac  +  16c2-24&c. 

12.  49  x"  +  56  a;?/  -  70  xz  -  AO  yz  +  16  y^ -^  25  zK 

13.  16a2-40a6  +  24ac-306c  +  256'^  +  9c2. 

14.  16aj2/  +  40a^/  +  36a;y  +  60ar*2/^H-73iiY. 

15.  a}'x^  -  6  a^V  _^  27  a^V  -  54  aV  +  81  aV^. 

16.  9aj2-54a^  +  l-6ic  +  729a;«  +  162a;^ 

17.  l  +  4a  +  20a^-16a''4-16a^ 

18.  a«~4a7  +  10a«-20o^  +  25a*-24a3  +  16a«. 

19.  25x*-{-10x^-^l  +  4:X-\-20a^  +  24a^  +  16a^, 

20.  a26V  +  8  a^ftV  +  4  a^ftV  _  4  a^ftV  +  4  a*6V  -  4  a^b*c^. 

21.  40  a»  +  25  ai«  +  4  a*  + 12  a*  +  25  a«  + 44  a^  + 46  a8. 

22.  36a^-84a:3_47a;4_^220a;«~62a^-144a;^  +  81ic». 

„„    a2  ,  4a6  ^4^2  ,  ac  ,  3  6c  ,  9c2 
^'-    9+"l5"+25  +  2+"5-^16' 


224  ELEMENTARY  ALGEBRA 


_.     d^  ,  ah  ,  2ac  .   1/  .   he  ,    c^ 

24.    — h  — — • 

9       6       15  ^16^10     25 

„^    4a2  ,     ,    ,  952     lOac     Qhc  ,  IGc^ 
26.   _  +  a&  +  ^-3^-^  +  -^. 

„^    4a2       3  ,  241  a^     5a^  ,  25a« 


9  144         4     '    36 

12     25_24     16^ 

X  X^  X^         iC'' 

a      a^      a'^      a*      cr      a" 

9a'      Sa'     25     56     25  6^ 
*   25  6^      5  6       4       a^        tt*  ' 

31.   ?|  +  ?5^69  +  36a;  +  36aj2. 

or       X 

Find  to  three  terms  the  approximate  square  roots  of  ; 
32.    1  +  x.  33.   1  — a;  34.    l+2a.  35.   9  +  4  71. 


233.  Tlie  square  root  of  arithmetical  numbers  can  be  found  by 
a  method  very  similar  to  the  one  used  for  algebraic  expressions. 

Since  the  square  root  of  100  is  10;  of  10,000  is  100;  of  1,000,000  is 
1000,  etc.,  the  integral  part  of  the  square  root  of  a  number  less  than  100 
has  one  figure,  of  a  number  between  100  and  10,000,  two  figures,  etc. 
Hence  if  we  divide  the  digits  of  the  number  into  groups,  beginning  at  the 
units,  and  each  group  contains  two  digits  (except  the  last  which  may 
contain  one  or  two),  then  the  number  of  groups  is  equal  to  the  number 
of  digits  in  the  square  root,  and  the  square  root  of  the  greatest  square  in 
the  first  group  is  the  first  digit  in  the  root.  Thus  the  square  root  of  96'04' 
consists  of  two  digits,  the  first  of  which  is  9  ;  the  square  root  of  21'06'81 
has  three  digits,  the  first  of  which  is  4. 


EVOLUTION 


225 


Ex.  1.     Find  the  square  root  of  7744. 

From  the  preceding  explanation  it  follows  that  the  root  has  two  digits, 
the  first  of  which  is  8.  Hence  the  root  is  80  plus  an  unknown  number, 
and  we  may  apply  the  method  used' in  algebraic  process. 

A  comparison  of  the  algebraical  and  arithmetical  method  given  below 
will  show  the  identity  of  the  methods. 

di  +  2  a&  +  &2  [q  +  ft  7744 1  80  +  8 

6400 
160  +  8 


rt^ 


2a+6 


2  a6  +  62 
2  a6  +  62 


168 


1344 
1344 


Explanation.  Since  a  =  80,  a^  =  6400,  and  the  first  remainder  is  1344. 
The  trial  divisor  2a  =  160.  Therefore  6  =  8,  and  the  complete  divisor 
is  168. 

As  8  X  168  =  1344,  the  square  root  of  7744  equals  88. 

Ex.  2.    Find  the  square  root  of  524,176. 

a        b      c 

52'iV76 1  700  +  20  +  4 

a^=  49  00  00 

2  a  +  6  =  1400  +  20  =  1420 1  3  41  76 

I  2  84  00 

2(a  +  6)  +  c  =  1440  +    4  =  1444 


57  76 
57  76 


234.  In  marking  off  groups  in  a  number  which  has  decimal 
places,  we  must  begin  at  the  decimal  point,  and  if  the  right- 
hand  group  contains  only  one  digit,  annex  a  cipher. 

Thus  the  groups  in  .0961  are  '.09'61.  The  group  of  16724.1  are 
1'67'24.10. 

Ex.  3.    Find  the  square  root  of  6.7  to  three  decimal  places. 

6'.  70      1  2.588 

_4 

45 


2  70 

2   25 


508 


4500 
4064 


6168 


43600 

41344 

2256 


226  ELEMENTARY  ALGEBRA 

235.  Roots  of  common  fractions  are  extracted  either  by  divid- 
ing the  root  of  the  numerator  by  the  root  of  the  denominator, 
or  by  transforming  the  common  fraction  into  a  decimal. 

.        V|  =  ±|;  V|  =  V.4. 

EXERCISE  91 

Extract  the  square  root  of : 


1.    1024. 

9.    27,889. 

18.   3,294,225. 

2.   5625. 

10.   75,076. 

19.    10,227,204. 

3.  4761. 

4.  8836. 

5.  1369. 

6.  6724. 

11.  57,121. 

12.  772.84. 

13.  328,329. 

14.  390,625. 

15.  74.1321. 

20.  17,850,625. 

21.  33,790,969. 

22.  30J. 

23.  AV4- 

7.   9604. 

16.   6037.29. 

24.    l^V 

8.    15,129. 

17.   857,476. 

25.    ^^i^. 

Find  to  three  decimal  places  the  square  : 

roots  of  the  follow- 

ing  numbers: 

26.    8. 

28. 

11.5.                 30. 

31^0 

32.   |. 

27.    10. 

29. 

147.6.               31. 

4i. 

33.    |. 

236.  The  fourth  root  of  an  algebraic  expression  or  arith- 
metical number  is  obtained  by  extracting  the  square  root  of  the 
square  root  of  the  given  quantity.  Similarly  the  eighth  root 
may  be  obtained  by  three  successive  extractions  of  the  square 

root.  » 

EXERCISE  92 

Extract  the  fourth  root  of : 

1.  a8-4a«  +  6a4-4a2  +  l. 

2.  m*  +  Sm^y-\-24:7nY  +  32mf-\-l^y*. 

3.  Sla^-tlOSa^y'-h54.xY+12xy  +  y^, 


EVOLUTION 


227 


4.   16  a«  -  96  a%^  +  216  a'W  -  216  a^b^  +  81  W^. 
6.   279,841.  7.   614,656. 

6.   456,976.  8.    1,874,161. 

(For  Cube  Roots  see  Appendix  IV.) 


REVIEW  EXERCISE  IV 
1.   Expand  (3a-6+c)^ 


2.   Find  the  value  of 


A  +  tJ  —  A , 

_7    1  1   _l_    1 

12         rs  ^  FB" 


3.  Simplify  ^  + 2!^^ -£^. 

4.  Find  the  mean  proportional  between 


a^  +  b^ 


and  a*  —  b\ 


(n2 7)2\2  a—  b 

5,  Find  the  third  proportional  to  ^^ v  and 

^    ^  (a  +  6)*  a  +  6 

6.  Simplify  [(a  -  6)*  +  (a  +  &)']  [(a  -  6)*  -  (a  +  6)"]. 
Solve  the  following  equations : 

7.   ]  a;:  «  =  14:  98, 

I  a; :  7  =  12  :  42. 

x  —  y-^z  —  u=—2f 
2x-\-y  —  3z-\-u=—l, 
\4:X  —  2y  —  z  —  u=—7, 
l5a;4-22/  +  22;H-w  =  19. 

(X     y-l_ 


8.   \ 


9. 


8 


If, 


-  _  y  ~^^  _  .v  +  16 

3  6      ■"      6     ' 


228 


ELEMENTARY  ALGEBRA 


X  —  2y~3z-{-u-{-v=—  2j 
2x-3u  +  2z-4.v  +  y=-22, 
2  y-Qz- 2  u  +  3x-v=- 17, 
5z  —  2v  +  3u  —  4:X  —  y  =  10j 
2v  —  4:U-\-3y  —  z-\-x=:—  4:. 

a"  -^ay  =  a*. 

12.  Find  the  numerical  value  of 

32  _  14 .  33  _  13  .  32  _  11 .  33  _^  25  .  3^  +  15  .  3\ 

13.  Find  the  value  of  (6000  +  7)^ 

14.  Find  the  9th  root  of  2^a''b^  («  ■^}>f. 


10. 


11. 


Extract  the  square  root  of  the  following  expressions ; 

15.    ^o?-ah  +  -V>-  ^^  +  tV  ^^  -  - 


^6c+||c^. 


16. 


x'^  +  10  x'z  4-  25  z^ 

^^     m^4-2m-f-l     2(m^  +  3m  +  2)  ^  3m^+12m  +  10 
d^  o?  a* 

2(m^  4-  5  m  +  6)      m^  +  6  m  +  9 
■^  a^  "^  a«  * 

18.  Solve  the  equations  : 

(a)  (-2)^  =  - 32. 
(5)   (_2)=»  =  64. 
(c)    (-3)^  =  -27.^ 

19.  Solve  the  following  system : 

{x'.y:z'.u  =  2x3'.4:'.5, 
\3x  —  2y  +  4.z  —  3u  =  l. 

20.  Eeduce  to  lowest  terms     ^    „  ^ — • 

7  a^- 12  a; +  5 


EVOLUTION  229 

.21.    Beduce  to  lowest  terms  «^  +  6^  +  c^  4-  2  a&  +  2  ac  +  2  6c  ^ 

22.  Simplify  {-V{a^"')f\ 
Solve  the  equations : 

23.  ^^-^(^^^-1^2:7. 

24     ^^  -^  ^"'^  ,      ^^    ^     3a;P 
2a;-4        a;  -  2     2a;-2* 


CHAPTER   XIV 

THE   THEORY   OF   EXPONENTS 

237.  The  following  four  fundamental  laws  for  positive  integral 
exponents  have  been  developed  in  preceding  chapters : 

I.  a'"  '  a"  =  a'"+". 

II.  a"'  -i-a"  =  a'"-"f  provided  m  >  n.* 

III.  (a"')"  =  a"'". 

IV.  (ab)'"  =  a"'  •  b"'. 

The  first  of  these  laws  is  the  direct  consequence  of  the  defi- 
nition of  power,  while  the  second  and  third  are  consequences 
of  the  first. 

FRACTIONAL   AND  NEGATIVE   EXPONENTS 

238.  Fractional  and  negative  exponents,  such  as  2 3,  4"^,  have 
no  meaning  according  to  the  original  definition  of  power,  and 
we  may  choose  for  such  symbols  any  definition  that  is  conven- 
ient for  other  work. 

It  is,  however,  very  important  that  all  exponents  should  be 
governed  by  the  same  laws ;  hence,  instead  of  giving  a  formal 
definition  of  fractional  and  negative  exponents,  we  let  these 
quantities  be  what  they  must  be  if  the  exponent  law  of  mul- 
tiplication is  generally  true. 

239.  We  assume,  therefore,  that  a"^  -  a''  =  «*"+",  for  all  values 
of  m  and  n.     Then  the  law  of  involution,  (a"*)"  =  a"*",  must  be 

*  The  symbol  >  means  "is  greater  than";  similarly  <  means  "is 
smaller  than." 

239 


THE    THEOBY    OF   EXPONENTS  231 

true  for  positive  integral  values  of  n,  since  the  raising  to  a 
positive  integral  power  is  only  a  repeated  multiplication. 
Assuming  these  two  laws,  we  try  to  discover  the  meaning  of 

8^,  a",  4  ^,  a  " ,  etc.  In  every  case  we  let  the  unknown  quantity 
equal  x,  and  apply  to  both  members  of  the  equation  that  opera- 
tion which  makes  the  negative,  fractional,  or  zero  exponent 
disappear. 

240.    To  find  the  meaning  of  a  fractional  exponent ;  e.g.  a\ 

Let  X  =  aK 

The  operation  which  makes  the  fractional  exponent  disappear 
is  evidently  the  raising  of  both  members  to  the  third  power. 

Hence  x^  =  (a^y. 

Or  x?  —  a. 

Therefore  x  =  ^a. 

p 
Similarly,  to  find  a  meaning  for  a', 

p 
we  let  X  =  a*. 

Raising  both  members  to  the  qth  power,  x^  =  a^. 
Taking  the  qth  root  of  both  members,  x  =  -\/a^, 
p 
or  a^  =  -Vo^. 

p 
Hence  we  define  a^  to  be  the  qth  root  of  a'. 


EXERCISE  93 
Find  the  values  of : 

1.  si 

5.  9l             9.   lel 

13.    (-32)i 

2.    4i 

6.    27l           10.    ll 

14.  (i)i 

3.    9k 

7.   32l           11.    Ol 

15.    (^V)^. 

4.    64i 

8.    125l         12.    (-8)*. 

16.    (i)k 

17. 

(a'-^2ab-^b^)k              18.    (a3- 

.3a'-^3a-l)k 

282  ELEMENTARY  ALGEBRA 

Write  the  following  expressions  as  radicals : 


19.    ai 

23. 

(-1- 

27. 

3 

20.    x\ 

24. 

4 

28. 

TO 

21.    m^*. 

25. 

m 
2 

29. 

22.    (ab)^. 

26. 

2". 

30. 

a™"". 

Express  with,  fractional 

[  exponents : 

31.    ^^. 

35. 

^3. 

39. 

^a-Vh. 

32.    -V^\ 

36. 

V^\ 

40. 

^a.Vb-</c. 

33.    -\/c^. 

37. 

'W. 

41. 

W. 

34.    Va. 

38. 

</a5. 

42. 

v'a»'+«. 

Find  the  values  of : 

43.    ^4. 

46. 

^2. 

49. 

■^2. 

44.    ^4. 

47. 

U 

50. 

</3. 

45.    </2. 

48. 

^\- 

241.   To  find  the  meaning 

of  zero  exponent,  e.g.  a^. 

Let 

aj=a« 

The  operation  which  makes  the  zero  exponent  disappear  is 
evidently  a  multiplication  by  any  power  of  a,  e.g.  al 

a;  =  -5  =  1. 

or  a«  =  l. 

Therefore  the  zero  power  of  any  number  is  equal  to  unity. 

Note.     If,  however,  the  base  is  zero,  —^  is  indeterminate  ;  hence  0"  is 
indeterminate. 


THE   THEORY  OF  EXPONENTS  283 

242.   To  find  the  meaning  of  a  negative  exponent,  e.g.  a~\ 
Let  .  a;  =  a~". 

Multiplying  both  numbers  by  a",  a"«  =  a®. 


Or 

a"a;  =  l. 

Hence 

1 

Therefore 

"-■4. 

243.  Factors  may  he  transferred  from  the  numerator  to  the 
denominator  of  a  fraction,  or  vice  versa,  by  changing  the  sign  of 
the  exponent.  -i 


a-"      1 


Note.  The  fact  that  a''  =  1  sometimes  appears  peculiar  to  beginners. 
It  loses  its  singularity  if  we  consider  the  following  equations,  in  which 
each  is  obtained  from  the  preceding  one  by  dividing  both  numbers  by  a. 


a^=l .  a  '  a  '  a 

a^  =  l'  a  -a 

«!=!.« 

aO  =  l 

a 

a-2  =  ietxj. 

EXERCISE  94 

ind  the  values  of : 

1.   4-1                 3. 

1-^                    5.    l-l« 

7.    3« 

2.    2-*.                 4. 

100«.              6.   2-^ 

8.  (4)*. 

234 

ELEMENTARY  ALGEBRA 

•4- 

14.  25-i 

15.  32-i 

16.  625-1 

17.  1    .            . 

19. 
20. 
21. 
22. 

a/10. 
-^32. 

12.    (f)-l 

36-^ 

23. 

8^  .  4-1 

13.    8-i 

18.    0-^ 

24. 

8^.4-i 

Express 

with 

positive  exponents : 

• 

25.    a-\ 

30      -^«"' 

33. 

3  a-'b-' 

26.    2a-^ 

30-         js 

6-' ad-'' 

27.  5a-V. 

28.  7a-»6- 

"c^ 

-■  W- 

34. 

9  a^&-^ 
4a-W 

29.    ^ -''''- 

-1 

32.    ^^~'^. 

Write  without  denominators : 

35.  12a^'.  37.    ^^.  39     ^ 

36.  ^.  38.    '^^'  40. 


o/. 

a^fz' 

38. 

a-'b 
xy* 

signs  and 

47. 

3 
2  m  ''. 

48. 

1 
2  - 

49. 

1 

2 
6  ^ 

50. 

2 

6^2  x?/^  eabc 

Write  with  radical  signs  and  positive  exponents : 

41.  a*. 

42.  a~K 

43.  5ai 

44.  5a~^. 

45.  (5a)-i 

n 

46.  2m~». 


51. 

1 
m  * 
3 

52. 

1 

a"» 

53. 

a* 

2-' 6-* 

64. 

-^a--^a-  </a. 

THE  THEOBY  OF  EXPONENTS  235 

THE  LAWS   FOR  NEGATIVE   AND   FRACTIONAL 
EXPONENTS 

244.  Exponent  latv  of  division  for  any  values  of  m  and  n. 
To  prove  a"»  -=-  a"  =  a""-"  for  any  value  of  m  and  n. 

^  =  a"*  X  —  =  a"» .  a-«  (§  243) 

=  a—".  (§  239) 

Hence  the  law  is  true  for  any  values  of  m  and  n. 

245.  Exponent  law  of  involution  for  any  values  ofm  and  n. 

To  prove  (a"*)**  =  a"***  for  any  values  of  m  and  n. 

Case  1.     Let  m  have  any  value,  and  w  be  a  positive  integer.     This  was 
proved  in  a  preceding  chapter  (§  239). 

P 
Case  2.     Let  m  have  any  value,  and  w  be  a  positive  fraction  -• 

p        

{a^y  =-^(a^)P  (§240) 

=  \^a^  •  (Case  1) 

mp 

=  a"^.  (§  240) 

Case  3.    Let  m  be  any  number  and  n  be  negative. 

1  1        __^- 


(a-)- 


(«"')'*     a^ 


246.   Ill  a  similar  manner  it  can  be  proved  that  the  lata 

(aft)"*  =  a'"^'"  IS  true  for  fractional  and  negative  exp>onents. 

P 
Case  1.    Let  the  exponent  be  -  when  p  and  q  represent  positive  integers. 

p     p 
Then  (a?  •  6? )?  =  aPbP  =  {aby. 

*  E      P  E 

Hence  ai  -b^  =  (aby. 

Case  2.     Let  the  exponent  be  —  n,  where  n  represents  any  number ; 
then  -.  ■, 

(aft)-"  =  -4^  =  -^  =  a-^b-\ 
{aby     a^b*" 


236  ELEMENTARY  ALGEBRA 

Hence  the  four  laws  of  exponents  are  true  for  any  value  of  the 
exponents,  and  we  have,  in  general,* 

Fractional  and   negative   exponents  are  treated  by  the  same 
metJiods  as  x^^^^^^^'^^  integral  exponents. 

247.  Examples  relating  to  roots  can  be  reduced  to  examples  con- 
taining fractional  exponents, 

Ex.  1.     (^ah~\)i  H-  Qah~i')\  -  ah~^  -f-  ah~^  =  arh^ 

=  ^  =  «/P 

Ex    2      /^^V^v^Y  =  h  «^&^Y=  ^^  ^^^^'^  =  ^^  ^^ 
Uv/«W       \3ahy      72Ua63     729  & 

248.  Expressions  containing  radicals  should  be  simplified  as 
follows : 

(a)  Write  all  radical  signs  as  fractional  exponents. 

(b)  Perform  the  operation  indicated. 

(c)  Remove  the  negative  exponents. 

(d)  If  required,  remove  the  fractional  exponents. 

Note.    Negative  exponents  should  not  he  removed  until  all  operations 
of  multiplication^  division^  etc.,  are  performed. 

EXERCISE  95 

1.  a«.2a-^.3a-.2a-^  7.   a'^a-\ 

2.  6a^'2a-\  8.    12a-^-i-6a-\ 

3.  x^  '  x^  '  xK  9.  14  a~^  -i-  2  a~^. 

4.  5T.5^.5T^.5i  10.  4m2.3m~^^6m-*». 

5.  x^'2x-\  11.  (-^/sy. 

6.  2a-^-2a^'2a\  12.  V2  .  a/2  •  ^/2. 

*  Irrational  and  imaginary  exponents  cannot  be  considered  at  this  stage 
of  the  work. 


THE  THEORY  OF  EXPONENTS 


237 


13.  -s/2'^-s/J\ 

14.  </r--VT\ 

15.  -^3.  ^3 --^3. 

16.  2h-V2. 

17.  2.-v/2--</2^. 

18.  -^7  . -^r  .  Vt. 

19.  ^4.^8. 
Hint,  4  =  2\ 

20.  ^4.^512. 

21.  ^8T.^27. 

22.  ^32.^/128. 

23.  (a;^)l 

24.  (-v/^)! 

25.  (^p)l 

26.  Vx  '  -y/x  •  Vx, 

40.  (Va)i 

41.  -^ai 

42.  -v/^. 

43.  -V^. 


27.  -\/a3 . -v^a  . -v^^ 

28.  ^a^.^a^.^a^ 

29.  ^a^ .  Va  .  ^  .  ^/'^. 

30.  -v^'125  .  a/5  .  ^25  .  ^/5. 

31.  ^^H--^^. 

32.  ^/^-ir^/'^. 

33.  ^--^a«. 


1+2 


34. 
35. 

36.  Va  •  Va  -5-  Va. 

37.  \/«-^(-\/^.-v/^). 


38. 


39. 


V2.-^4 
a/32 


44.  V^^. 

45.  a/;j7^. 

46.  (Vxy)^ 

47.  (V^V^)-2. 


^^-</^ 


48. 


49. 


/27a-«\-^ 
1^64  6-V 


54 


^ad^     -y/as^ 


•sJo^X      WOJ^X 

55.  '^l(y^)-'^^{^l/^-^ 

56.  ^(a^.&^)-^- 

57.  (v/o^.a/o^)"*. 


58 


238 


59. 


^~^y^~^ 


X^y-'i 


ELEMENTARY  ALGEBRA 

3«+l    ^    (32^)n-l 


x~^y~^^^ 


oiy^if 


-Cy"- 


63. 


64. 


4«+i       3„2_i 


61.    "Vx-'y-h-'  -  A/aj-y-^-. 


62. 


■y/x  '  ic* 


^(«.t)i 


®^-    \(a-6)-'  ■   \(a-6)-^- 

66.  C/i^.^^.^i^. 

\     c/—  1/     a,—  1/     6/— 

S/a^  vic         va; 


249.  If  we  wish  to  arrange  terms  according  to  descending 
powers  of  x,  we  have  to  remember  that  the  term  which  does 
not  contain  x  may  be  considered  as  a  term  containing  x^.  The 
powers  of  x  arranged  are  : 

Ex.  1.     Multiply  3x-'^  +  x-5  by  2a;-l. 

Arrange  in  descending  powers  of  x. 


Check. 

If 

x  =  l 

x-6  +  Sx-^ 

=  -1 

2a;-l 

=  +1 

2x2- 10a;  +    6 

—      a;  +    5  - 

-.Sx-i 

2x2-  llx+  11  - 

-3x-i 

=  -1 

Ex.  2.     Divide 


at  _  6  a&3  +  9  a^ft'-  _  4  5I 

J-Sah^  +  2h^ 

at  _  3  a6^  +  2  J^t 

af  _  3  a^5^  -  2  6^ 

-  3  a&^  +  7  a"3&*  -  4  6^ 

-  2  a^&*  +  6  a^?> 

-4&3 

-  2  a3'6^  +  6  a^6  -  4  63 


THE  THEORY  OF  EXPONENTS 


239 


Ex.  3.     Find  the  square  root  of 

ar  —  2x  V a  -\-3a ^^—  +  — • 

X  x^ 

Write  with  fractional  and  negative  exponents : 


(C2     _  2  a^: 


x-\-3a-2 a"2a;-i  +  a'^x-^  \  x  —  a^  +  ax-'^ 


2x-a^ 


2a^x  +  Sa 
2a^x+     a 


2x-2a^  +  aaj-i 


2  a  -  2  a^x-^  +  a'^x-^ 
2a -2  a^x-i  +  a^x-a 


This  answer  may  be  written  without  negative  and  fractional  exponents  i 

X  —  V  a  +  — 

X 

EXERCISE  96 
Perform  the  operations  indicated : 

1.  (3a-^-4a-4  +  5a-3)(2a-2-a-i). 

2.  {2a^-Sx-^-^2x-S)(2x-l). 

3.  («-"*  +  x"'  4- 1)  (a?""*  4-  a;"*  —  1). 

4.  (3a4-5Va  +  3)(3a-5Va  +  3). 
6.    (a^  +  2a-a^-2)(2a-l). 

6.  (Va4-V6  +  l)(Va  +  V&-l). 

7.  (^/a2  4-2-^a6-^P)(^/^-3\/^  +  2-v^P). 

8.  (a"*  — 2  +  a-'«-a-2«)(a'»  +  a-'»  +  l). 

9.  (4:2x-10x^y^-12y)-i-(7x^  +  Sy^), 

10.  (2a^4-18-3a^-7a)-;-(2a^  +  3). 

11.  (4a2  +  30a-i-9-25a-2)^(2  +  3a-i-5a-2). 

12.  (12^2_28-^a  +  15)^(6^a-5). 

13.  (a^6)-^(^/a--^6). 


240  ELEMENTARY  ALGEBRA 

14.  (4-v/a-f3  +  ^^*)--(-v'a2H-3-2^a). 

15.  (a^»*-6a)'"-19  +  84a;-'»).-^(l-7x-'"). 


17.    V9a-2  +  24a-^  +  46  +  40a-f 25a2. 


18.  Va;2_6x^  +  13a;-12x2  +  4. 

19.  Va;-2-2x  +  a^-2x-i  +  3. 

20.  Vg a;2 - 12 V^  + 34 aj- 20 Vx  +  25. 
21. .  (49  a^  -  42  Vo^  +  37  a6  - 12  Vc#  +  4  d*^i 

22 .  (2  aj"*  +  2  X-"*  +  3  +  X-'"  +  a;-^'^) i 

23.  (2aa;  +  16^9^V-8A/72S^)*. 

24.  (l  +  a  +  2V^  +  -v^^  +  2va"2  +  2-J/<)i 

25  1^  4a        2V^     19      3-^5      ^ft'n^ 

*  \^-Vh'       </b       12     2Va      4a;  ' 

26.  (3+V5  +  2V6)(V5  +  V6). 

27.  (24-3V3  +  4V6)(2V3  +  3V6). 

28.  (4V54-5V6  +  4)(5-|-6V6). 

30.  (VS  +  V^  +  l)(^  +  ^-l). 

31.  (2-V^  +  i»)'. 

32.  (l-Va  +  a)«. 

33.  (l-4Va4-6a-4aVa  +  a^*. 
4Va  4      415  5      1616^      2b^\  ,  /2V^     3 


34.    r-7^-^— ^±ii^H-±^±:i-— ^1^   -   ^^f^^-^  + 


3&       3     36va      18a      Vo^/     \    ^        2     3Va 


H 


THE  THEORY  OF  EXPONENTS 
rind  by  inspection  the  values  of  the  following : 


241 


35. 
36. 
37. 
38. 
39. 
40. 
41. 
42. 
43. 
44. 
45. 
46. 
47. 
48. 


a-\-a-y. 
x-x-y. 

x"2-9)(a;"^  +  9). 
x^  +  lf. 

a^+iy. 

«^  +  ^^)';  ^       ^  57.    V^  +  2mi  +  l. 

ah^  -  ah^)  {ah^  +  ah^).  ,~ — 

58.    '^a^-\-2am  +  hK 

-  7  ¥)  (a*  -  3  ¥y  ^9-    (9  «"'  -  12  ^'^^"^  +  4  6"^)^. 

a-«_6-«)2..  60.    {x'  +  2  +  x-^)'\ 

a-" +  2  a"/.      '  61.    (a^  +  3a^  +  3a^  +  l)l 


49. 

(a  +  l  +  a-y. 

50. 

(m"  +  2  —  m-«)2. 

51. 

(a;-i  +  2/-^  +  ^-i)2. 

52. 

(x-2-rO(^-^-r^).' 

53. 

(a-6)-(a>4-?>b. 

54. 

(a-h)^{a^-}fi). 

55. 

(a-3_6-3)^(«-i_^-i) 

56. 

(x  +  8)-(a^^  +  2). 

CHAPTER  XV 
RADICALS 

250.  A  radical  is  the  root  of  a  quantity,  indicated  by  a 
radical  sign. 

251.  The  radical  is  rational,  if  the  root  can  be  extracted 
exactly ;  irrational,  if  the  root  cannot  be  exactly  obtained. 
Irrational  quantities  are  frequently  called  surds. 

a/9,  (x  +  y)^  are  radicals. 

4^  =  2,  ^(a  +  h)^  are  rational. 

V2,  V4  tt  4-  &  are  irrational. 

252.  The  order  of  a  surd  is  indicated  by  the  index  of  the 
^^^^'  Va  is  of  the  second  order,  or  quadratic. 

\/2  is  of  the  third  order,  or  cubic. 

Vc  is  of  the  fourth  order  or  biquadratic. 

263.  A  mixed  surd  is  the  product  of  a  rational  factor  and  a 
surd  factor ;  as  3  Va,  a;V3.  The  rational  factor  of  a  mixed 
surd  is  called  the  coefficient  of  the  surd. 

An  entire  surd  is  one  whose  coefficient  is  unity;    as  V«, 

254.  Similar  surds  are  surds  which  contain  the  same  irrational 
factor.  ■      3v^  and  5aV2  are  similar. 

.3\/2  and  3\/3  are  dissimilar. 

255.  Conventional  restriction  of  the  signs  of  roots. 
All  even  roots  may  be  positive  or  negative, 

e.g.  V4  =  +  2  or  —  2. 

Hence  6Vi  +  2v'4  =  5(±  2) +2(±  2), 

242 


BADICALS  243 

which  results  in  four  values,  viz.  14,  6,  — 14,  or  —  6.  To  avoid 
this  ambiguity,  it  is  customary  in  elementary  algebra  to  restrict 
the  sign  of  a  root  to  the  prefixed  sign. 

Thus      "  5V4  +  2V4  =  7V4  =  14. 

5V2-V2  =  3V2. 

If  the  object  of  an  example,  however,  is  merely  an  evolution, 
the  complete  answer  is  usually  given ;  thus 


V^_4^  +  4  =  ±(x-2). 

256.  Since  radicals  can  be  ivritten  as  powers  with  fractional 
exponents,  all  examples  relating  to  radicals  may  he  solved  by  the 
methods  employed  for  fractional  exponents. 

Thus,  to  find  the  nth  root  of  a  product  ab  we  have 
1        11 
(a6)-  =  a"6"    (§246). 

I.e.  to  extract  the  root  of  a  product,  multiply  the  roots  of 
the  factors. 


TRANSFORMATION  OF  RADICALS 

257.  Simplification  of  Surds.  A  radical  is  simplified  when 
the  expression  under  the  radical  sign  is  integral,  and  contains 
no  factor  whose  power  is  equal  to  the  index. 


Ex.  1.    Simplify  ^2b  a^h. 

V26a^  =  a/25o*  •  Vfe  =  5 a'^y/h. 
Ex.  2.   Simplify  -s/i6. 

Ex.  3.    Simplify  ■\/Wc^^  - 


244 


ELEMENTARY  ALGEBRA 


258.  When  the  quantity  under  the  radical  sign  is  a  fraction,  we 
multiply  both  numerator  and  denominator  by  such  a  quantity 
as  will  make  the  denominator  a  perfect  power  of  the  same 
degree  as  the  surd. 

Ex.  4.    Simplify  VJ. 


Ex.5.    Simplify  ^1^. 


Simplify ; 


■y^    Sab'^ 


9  a^b    3  a&2 

EXERCISE  97 


'    2<  a^b^       Sab 


1.  V28. 

2.  V45. 

3.  Vl8. 

4.  V24. 

5.  V27. 

6.  V96. 

7.  V243. 

8.  V320. 


12.  ^2^3  a^b\ 

13.  3V8. 


23. 
24. 


-^648. 


14.  5^/SOo^. 

15.  8V757^. 


16.    eVlSOa^ftl 


17.    3Vl2a3. 


18.    7V48aicl 


19.    |V24al 


9.    V4()5a2. 


10.    V363^. 


11.    V24a«„ 


20.    #V27  6\ 

21, 

22.    ^• 


-^80. 


81. 


25. 

2V-48al 

26. 

7  V108  a'b*. 

27. 

5V320  a\ 

28. 

3  V16  a^6^ 

29. 
30. 

31. 

^^2n^3n+2^ 

32. 

V3(a  +  6)^ 

33.    V2(a2  + 2  a& +  ?>'). 

39.    VJ. 


34. 


35.  Won^  —  oc^. 

36.  (9a  +  18)^- 

37.  (64a«6V'^)i 


40. 


38.    Vi 


41.    V|. 


42.  V|. 

43.  V|. 

44.  VA- 

45.  VJ^. 


47. 

H'r 

48. 

W?- 

49. 

«^?- 

50. 

^• 

51. 

^. 

52. 

:\'< 

BADICAL8 

246 

54. 

60. 

2y'yi  8a^' 

55. 

61. 
62. 

Vsic^- 

56. 

57. 

3    /13  a' 
a^VlSx 

63. 

(«-)Vj?|- 

58. 

^- 

64. 

59. 

2  a  '/27  6* 
6   \  2a 

65. 

•e-- 

53.    V|. 

259.   An  imaginary  surd  can  be  simplified  in  precisely  the 
same  manner  as  a  real  surd  ;  thus 


V-9=3V 
Simplify : 

68.    V-196. 

ee.  V-¥- 

V^=fvr-a. 

66.    V— a-. 

-  SV-¥- 

71.    V-a"-2o6- 

67.    V-16m. 

-6^. 

260.   Reduction  of  a  surd  to  an  entire  surd. 
Ex.  1.    Express  4  aV5  as  an  entire  surd. 
4  a  V&  =  \/l6a2\/6  =  ^/Wofih. 


a  "  /2'*~^6""^^ 
Ex.  2.   Express  — \/ -—  as  an  entire  surd. 

2  6^    a^+i        ^     2'»2)«a"+i  ^2  a* 


246  ELEMENTARY  ALGEBRA 

EXERCISE  98 

Express  as  entire  surds : 


1. 

2V3. 

2. 

3-^2. 

3. 

S</-2, 

4. 

2^3. 

5. 

iV6. 

6. 

1^4. 

7. 

a'Vb. 

8. 

-v^^, 

9.    |Vl5a^  j^     a  +  &    /a -6. 

10      ^  '''^ 


a-\-b    ja_ 
a  —  b  ^a 

^5a«  15.    !^Vi'^^ 


^j     2a  3/276 


12.    ^ 


J4Z       17. 2^.^rz!i. 

n  ^ 


13.    ^VE  18.    ^ 


261.    Transformation  of  surds  to  surds  of  different  order. 
Ex.  1.     Transform  Va^6^  into  a  surd  of  the  20tli  order. 

Ex.  2.  Transform  V2,  V3,  and  V5  into  surds  of  the  same 
lowest  order.  _       1        «       i,  _ 

</5  =  5^  =  5T2  zzz  V125. 
Ex.  3.     Reduce  the  order  of  the  surd  -^/aP, 

Exponent  and  index  hear  the  same  relation  as  riumerator  and 
denominator  of  a  fraction ;  and  hence  both  may  be  multiplied  by 
the  same  number^  or  both  divided  by  the  same  number^  without 
chayiging  the  value  of  the  radical. 


RADICALS  247 


EXERCISE  99 

Eeduce  to  surds  of  the  6th  order : 

• 

1.  -y/x. 

2.  -y/ab. 

3.  J^.                ''    ^1^- 

^^                 6.    4. 

4.  ab. 

7.  %-i«. 

8.  ^^. 

Reduce  to  g 

mrds  of  the  12th  order : 

9.    -Va'bK 

11.    -y/a'b'c'.        13.    2. 

15.    -s/a~\ 

10.    -y/a^K 

12.    abc^.              14.    -s/I. 

16.    ~-v/?^. 

Express  as 
and  indices : 

surds  of  lowest  order  with  in 

tegral  exponents 

17.    %^. 

21.    ^/8.               25.    -^27  a'b^ 

-.  ^«r- 

18.    ^4. 

22.  v^a-^'.           26.    a/81  a^6^ 

23.  ^/a^y.           27.    %^. 

19.    v27. 

30.   ^'/25«^ 

20.    ^49. 

24.    A/32itV-      28.    %76^. 

Express  as  surds  of  the  same  lowest  order : 

31.  V2,  v'a.  37.    2%  3l 

32.  ^/a,  a/6.  38.    -y/^,  -y/a',  V^\ 

33.  a/4,  a^.  39.    a/3,  a/3,  a/4. 

34.  a/7,  a/9.  40.    a/8,  a/8,  a^. 

35.  a/2,  -v^3.  41.    A^2,  a/8,  v4. 


36.    a/2,  a/5.  42.  Va^^^d^  Va^fe^  Va^ 

Arrange  in  order  of  magnitude : 

43.  a/2,  -yjl.  47.  a/10,  A/90i. 

44.  -^4,    ^6.  48.  V|,   a/|,  ^. 

45.  a/7,  a/2.  49.  5a/2,  4^4. 

46.  a/5,  ^n,  A/i24.  50.  4 a/8,  2v'9,  3 a/15. 


248 


ELEMENTARY  ALGEBBA 


ADDITION  AND   SUBTRACTION  OF   RADICALS 

262.  To  add  or  subtract  surds,  reduce  them  to  their  simplest 
form.  If  the  resulting  s^irds  are  similar,  add  them  like  similar 
terms  (i.e.  add  their  coefficients) ;  if  dissimilar,  connect  theyn  by 
their  proper  signs. 


Ex.1.    Simplify  Vi  4- 3  Vl8  -  2 V50. 

V}  +  3Vl8-2V50  =  ^V2  +  9V2-  10\/2 


iV2. 


Ex.  2.    Simplify  -\/a' 


o    0    , 


3/27  cc 


>ic-2       ^  2/3  X  y  \  X       y) 

Ex.  3 .    Simplify  Vf  -  ^i  +  V72  -  4 V^  +  ^/i6. 

V|  ~  v^J  +  V72  -  4VJ  +  v^  =  f  V2  -  I  v^2  +  6  V2  -f  V3  +  2  v^2 

=  VV2  +  |V^-fV3. 

EXERCISE   100 
Simplify  the  following  expressions : 

9.    2 Vi50  -  4V54  4- 6 V24. 

10.  2  V32  +  iv'50 -h  |V72. 

11.  f V44  +  4 V99  +  5  VTfe. 

12.  2Vl25-fV45  4-4V20. 

13.  -^i6-v'54  +  ^2. 

14.  2^/i6  +  3^/250  -  4-v/l28. 

15.  4^54  + 5^250 +  2\/l6. 


1.  V8-V32-V72. 

2.  V18-V32-V50. 

3.  Vl28-V32-Vi8. 

4.  V45-V245  4-VI80. 

5.  V80-V20-V5. 

6.  V63+ a/700 -V175. 

7.  3V20  +  5V45-4V80. 

8.  3V8  +  2V32  +  4V72. 


16.   3V250-2V686+2V1458. 


17.  4V^  +  3V62a;-2aVx. 

18.  3Va^-3aVaP-3a&Va&. 


BADICALS 


249 


19.    V34-V14-V27.  „,     _/^  .  ./3a 


20.    V5  -V20+V4-. 


21.    V6a4-\/^-V24a. 


^'  22.  Vi20+V|  +  iV|. 

23.    V32-f  3Vi  +  4V72. 

„.        lab     ^  lab  ,     lab  ,     I  ab 

25.  i-Vf  +  iVio  +  sVI-Vip'. 

26.  3v/;^-4j;^  +  -|^V84^. 

Via;        \3a;     21a; 

27.  VH  +  5 V5|  -  3V8|. 

28.  3V3l-5V9f-7Vl25. 

29.  1^^-1^141  +  1^6. 

30.  ^^^  +  ^^.»r._4^^. 


31. 


2  a:  +  Vl6  a;  +  V50^  4- V2^. 

32.  A/T28^+V72a6-V50a6+^/^r^. 

33.  A/8r^^  +  2A/l6^^-A/256^. 

34.  -</(?-^a4-V(a4-6ya. 

35.  V4  +  4a;2-fV9  +  9«2-5vTT^. 

36.  ^A  +  V¥x-\-V^. 

37.  -v/a¥3  +  A/a^-V4^. 
/aV  .     la^coif 


,        la*c 


39.  -s/a''x'-\-V27a'x-^125a:'x, 

40.  ^^  +  #. 


41 


94-V^^-V^^25. 


42.    V-12a2  +  V-75a2-V-48a2. 


250  ELEMENTARY  ALGEBRA 


MULTIPLICATION   OF   RADICALS 

263.  Surds  of  the  same  order  are  multiplied  by  multiplying 
the  product  of  the  coefficients  by  the  product  of  the  irrational 
factors,  for  aVx  •  bVy  —  ab^xy. 

Dissimilar  surds  are  reduced  to  surds  of  the  same  order,  and 
then  multiplied. 


Ex.  1.    Multiply  3V25  2/'  by  5V50/. 

3\/257  .  5\/50l^  =  15v^52.2'52.2/*  =  75  y^/W^. 

Ex.  2.    Multiply  V2  by  3\/4. 

V2  .  3 v/4  =  x/23  .  3^/42  =  ^23  .^W  =  3 v^  =  6\/2. 


Ex. 

3.    Multiply 

5V 

7  -  2  V5  by 

3V7  +  10V5.     . 

i 

5V7 

-      2\/5 

BV7 

+  loVs 

105 

-    6\/35 
+  50V36 

-IOC 

1 

105  +  44V35 

-IOC 

l  =  5  +  44V35 

Simplify : 

EXERCISE 

101 

1. 

V7.- 

\/42. 

11. 

7V5  X  •  V^. 

2. 

Vio- 

V5. 

12. 

3V^  .  Vt^. 

3. 

2V3. 

.Vl2. 

13. 

4-v'2  d^ .  -^4  a. 

4. 

2V28 

:.V7. 

14. 

3  a^a^ .  -Va'. 

5. 
6. 

7. 

^4.- 
-^25. 
3</49 

^2. 

15. 
16. 

6  x^x^y  •  A/a^2/- 

8. 
9. 

3^2. 
aVa 

.^4. 
■  x^fx. 

17. 

v-«-V^f- 

10. 

5Vaa 

;.3V5 

'^X. 

18. 

(V3+V4+V5)V5. 

BADICALS  251 

19.  (2V3  +  3V4  +  4V5)V3.     21.    (a-\-Vb)(a-Vb). 

20.  (3V5  +  2V3+V6)V6.        22.    ( Va  +  V6)  ( Va  -  V6). 

23.  (V3a  +  V26)(V3^-V2^). 

24.  (3V5H-2Vil)(3V5-2VlT). 

25.  {■y/x  +  y-{-Vy)(-Vx-{-y-Vy). 

26.  {■\/x-\-^x  —  y){-y/x  —  -\/x  —  y). 

27.  (  V9  a;  +  5  +  3  Va;)  ( V9  a;  +  5  -  3 V^). 

29.  (V^+V6)l  40.  (^/3-^2)«. 

30.  (V3  +  V2)2.  41.  (Va  +  V6)(a  +  5-VS). 

31.  (1+V2f.  42.  (Va  +  V6+V^)'. 

32.  (V6-V5)2.  43.  (V2+V5-Vl0)2. 

33.  ( V^T^  +  ^x-yf.  44.  Va^  •  VaF"  •  Va. 

34.  (a  +  Vn=^^)2.  45.  ^5/^^.-^^^.^. 

35.  (-1+V3)l  _      46.  ^T.^|.^|. 

36.  (3V2-2V5/.  47^  ^^e^^.-Vb^^=^\ 

37.  /'V^_-l-Y.  48.  ^^3  +  53 .  ^^3Zrp. 

/    /-         L  49.    (V7-V3)(V3-V2). 

^^'    \^y~^^^)'  50.    (3V2-2V3)(7V2+5V3) 

39.    (Va  +  V6)«.  51.    (5V3  4-V6)(5V2-2). 

52.  (2V6  +  5V3-7V2)(3V3-V2). 

53.  (3+V6+Vl5)(2  +  V6-VlO). 

54.  (^3  +  ^2)(2^9-3^4).    g^^    ^-.^/l 

5/-     5/-  *  ^a 

55.  va  •  va-  r—       p- 

m      s  n 
^^       3/-     6/-  58.    \--\/— 

56.  V c  •  vc  ^M     ^m 


252  ELEMENTARY  ALGEBRA 

59.  ^I-Ve.  64.  ^C'^/C'</10^. 

60.  ^f-Vj.  65.  (V6H-\/4)(V6--v/9). 

61.  Va-V6.  66.  (V5+-v/25)2. 

62.  ^/x-'-Vy.  67.  (V2+v'4)2. 

63.  ^2..^f. 

DIVISION  OF  RADICALS 

264.  Monomial  s^irds  of  the  same  order  may  be  divided  by 
multiplying  the  quotient  of  the  coefficients  by  the  quotient  of  the 

surd  factors.     E.g.  aVb  -f-  x-Vy  =  -\[— 

Since  surds  of  different  orders  can  be  reduced  to  surds  of  the 
same  order,  all  monomial  surds  may  be  divided  by  this  method. 

Ex.1.   4V48-f-3V6  =  |V8  =  |V2. 

Ex.  2.    ( V50  +  3 Vl2)  -  V2  =  VM  +3 V6  =  5  +  3  V6. 

265.  If,  however,  the  quotient  of  the  surds  is  a  fraction,  it  is 
more  convenient  to  multiply  dividend  and  divisor  by  a  factor 
which  makes  the  divisor  rational. 

This  method,  called  rationalizing  the  divisor,  is  illustrated  by 
the  following  examples : 

Ex.  1.    Divide  Vll  by  VT. 

In  order  to  make  the  divisor  (V?)  rational,  we  have  to  multiply 

by  >/7.  vn  ^  Vn  ^  V7  ^  V77  ^  lyy^^ 

Ex.  2.    Divide  AVSa  by  S^s/Tb'. 

The  rationalizing  factor  is  evidently  \/4  &  ;  hence, 


RADICALS  268 

Ex.  3.   Simplify  ^J^,- 

Ex.  4.   Divide  12V3  +  4V5  by  V8. 

Since  Vs  =  2\/2,  the  rationalizing  factor  is  V2, 

12\/3  +  4\/5  ^  12V3  +  4V5  ^  V2  _  12\/6  +  4VT0  ^  g^^      ^j^ 
V8  V8  \/2  4  ~ 

266.  To  show  that  expressions  with  rational  denominators 
are  simpler  than  those  with  irrational  denominators,  arith- 
metical problems  afford  the  best  illustrations.     To  find,  e.g. 

by  the  usual  arithmetical  method,  we  have 

V3     1.73205 

-p  ,  ..  .      T-P  1        V3     1.73205 

But  it  we  simpiiiy      —  =  -— -  =  — 

V3       3  3 

Either  quotient  equals  .57735.  Evidently,  however,  the 
division  by  3  is  much  easier  to  perform  than  the  division  by 
1.73205.  Hence  in  arithmetical  work  it  is  always  best  to 
rationalize  the  denominators  before  dividing. 

EXERCISE  102 
Simplify : 

1.  Vl2-hV6.  4.    ■\/Wx^-V2x. 

2.  VI8--V2.  5.    -s/Si-t-'s/S. 

3.  ■\/2^-f-V^.  6.    (4V6  +  4V2)-5-V2. 

7.    (34^  + 5^/16 -2^28) -^4. 

8-    ^^^-  11.    Vah^yf^. 


4-4 


9. 

^^ 1"*  12     ^2¥i  -  A /^- . 

10.    Vx'-y'-^Vx^.  '  ^  '  yi4:f 


254  ELEMENTARY  ALGEBRA 


13.     -^.  18.    ~^.  23.    ^a±^. 

Va  2V3  3^/1 +  a2 


14.  ^.  19.     -i?^.  24.    ^ 
\/a                               5V32  ■</a 

15.  -^.  20.     ^.  25.    -^ 

V3                          "v/tt  Vtt' 


16.  A:  21.    -^.  26.      "'-^'. 
V2                               -v/a»-»  Va-6 

17.  J,.  22.       ^  +  ^^   • 

Given  V2  =  1.41421,  V3  =  1.73205,  V5  =  2.23607,  and 
V6  =  2.44949,  find  to  four  decimal  places  the  numerical  values 
of: 

27.  -i=-  29.    A.  31.    -^. 

V2  V3  V8 

15  4 

28.  -^-  30.    -^.  32.    — ^. 

V3  V5  V50 

33.    -A-.  34.    -^.  35.    -^.  36.      ^^ 


V125  V6  .    V20  V45 

267.  Two  binomial  quadratic  surds  are  said  to  be  conjugate, 
if  they  differ  only  in  the  sign  which  connects  their  terms. 

Va  +  Vb  and  Va  —  Vb  are  conjugate  surds. 

268.  The  product  of  two  conjugate  binomial  surds  is  rational. 

(Va  +  Vb)(Va  -  y/b)  =  a  -  b. 

269.  To  rationalize  the  denominator  of  a  fraction  whose 
denominator  is  a  binomial  quadratic  surd,  multiply  numerator  and 
denominator  by  the  conjugate  surd  of  the  denominator. 


Ex.  1.     Simplify 


RADICALS  255 

2V3-V2 
V3-V2 


V3-V2        V3  -  \/2      Va  +  V2       3-2 


Ex.  2.     Simplify  ^~^^~^. 


Vx2-]      «  -  Vcc^  -  1     x-Vx2-l      x'^-2x^x-^-\  +  x^-l 


a;  +  Vx=^_l     a;  +  Va;2-l    ^^-Vx^-l  x"-^-(x-^  - 1) 


=  2x2-l-2iBVx2-l. 

Ex.  3.     Eind  the  numerical  value  of : 

V2  +  2 
2V2-I 

v^  +  2  _  V2  +  2     2\/2  +  1^6  +  5\^^  13.07105  _  ^  gg^g 
2V2-I     2\/2-l     2V2  4-1  ^  "^  *         * 

270.   If  the  denominator  is  a  trinomial  quadratic  surd,  two 
multiplications  are  necessary  to  rationalize  the  denominator. 

Ex.  4.     Eationalize  the  denominator  of : 

1+V2+V3 

I4.V2-V3* 

l.fV2  +  V3_   1+V2  +  V3        1  +  A/2  +  V3    ^6  +  2>/2  +  2V3+2V6 

l  +  V2-\/3      (l+V2)-\/3     (1  +  V2)+V3  (l+2V2-|-2)-3 

_  3  +  V2  +  \/3+V6  ^  j\/2  ^  3\/2  +  2+V6+2\/3 

V2  *  V2  2 

EXERCISE  103 
Rationalize  the  denominator  of : 
1.    _1_.  3.         ^^     .  5.       ^^ 


2+V2  4-2V3  7-3V2 

o  2  .       9V10  ^       IIV15 


2-V3  4  +  2V6  10-3V6 


^50  ELEMENTARY  ALGEBRA 


2-V2  j2     3V7-5V3         ^^ 


2  +  V2  3V7  +  5V3  Va?  +  Vy 

g^    3  4-vT  ,^3^   5V3-3V5         ^^     Vx-Vy 

3-V3' 

24 


10. 


11. 


22. 


23. 


V8-2 

V5-V3 
V3  +  V2" 

7V2-5V3 
4V3-3V2 

2 


V5-V3 

■y/x  +  Vy 

^^     7^/5  4- 5V7 

V7  +  V5 

19. 

5  +  -v^ 

1,.    V6-V3 

V3-V2 

20. 

c  +  dV^ 

16             ^ 

21. 

aa;  —  6a; 

«t  +  Va 

Va  +  V6 

27.    — 

2V2 

V0--4  *    i_^V2-V3 

2aa;  1 
28.  ^ 


V2cu»-fa;Va  V2  4-V3+V5 


24.    Va  +  6  +  Va  — &^  gg  I4-V2 

Va  +  6-Va-6  *   1+2V2  +  3V3* 

25. 


a^  1 


30. 


a  +  Va^-6^  2V1+1-V5 

26.    -r^ 31.  ^ 


3^-1  VV2-I 

Given  V2^  1.41421,  V3  =  1.73205,  and  V5  =  2.23607,  find 
to  four  places  of  decimals : 

32.    —^.  34.         ^    _■  36.       ^ 


I+V2  3-V5  2-V3 

83.    -i^.  35.    ^^-A  37.  1 


2-V3  V6+V3  V12-VS 


RADICALS  ^67 

INVOLUTION   AND  EVOLUTION   OF  RADICALS 

271.  Powers  and  roots  of  radicals  can  be  found  by  the  use 
of  fractional  exponents. 

Ex.  1.    Find  -y/^, 

Ex.  2.    Find  the  square  of  ^2^. 

(  \/2^)2  =  (2  x2)  ^  =  \/ra  =  X  ^/Ix, 

272.  A  case  which  frequently  occurs  is  expressed  by  the 
formula  Va"*  =  (-\/ay\     This  equation  is  correct  since  each 

m 

member  equals  a".  It  should  be  remembered,  however,  that 
the  signs  of  the  two  members  are  equal  only  if  they  are 
restricted  in  the  manner  of  Article  255.  If  the  radicals  are 
,taken  with  all  possible  signs,  Va"*  does  not  necessarily  equal 
(■y/a)"'j  as  illustrated  by  the  following  example : 

( Va)^  =  a,  by  definition  of  square  root, 

Ex.  3.   Find  {2^aF^y. 

(2  v^^^)*  =  2*  y/^^  =  16  a2a;6  ^/^2^, 

Ex.  4.   Find  -s/TU\ 

\/l4i8  =  (  y/Tuy  =  (  ±  12)3  =  J.  1728. 

EXERCISE  104 
Simplify : 

1.  (2V^)^  6.    (4a^^)«.  9.    -v^ivS. 

2.  (2Vxf,  ,  6.   ^/S^  10.   VvS. 

3.  {2Vx)\  7.    -v/125"^  11.    ■\^. 


4.    (2a's/af)\  8.    ^/W.  12. /V^/^?, 


258  ELEMENTARY  ALGEBRA 

SQUARE  ROOTS  OF  QUADRATIC   SURDS 

273.  A  quadratic  surd  cannot  he  equal  to  the  sum  of  a  rational 
expression  and  a  quadratic  surd. 

For,  if  possible,  let  Va  =  6  +  Vc. 

Squaring  both  members,        a  =  b^-\-2  bVc  +  c. 

Solving  for  Vc,  Vc  =     ~     ~   « 

Z  b 

That  is,  a  quadratic  surd  is  equal  to  a  rational  expression, 
which  is  impossible. 

274.  If  a  +  V6  =  jr  +  V/,  then  j  =  jr  and  6  =/. 

Suppose  a  were  not  equal  to  a;. 

Transposing  a,  Vb  =  a;  —  a  +  Vy, 

which  is  impossible.     (§  273.) 
Therefore  a  =  x,  and  hence  b  —  y. 

275.  To  find  the  square  root  of  a  binomial  surd. 
To  find  the  square  root  of  17  +  V240. 


Assume  Vl7  +  V240  =  V^  +  V^. 

Squaring  both  members,  17  +  V240  =  x  +  2^/xy  +  y. 
Then,  by  §  274,  x-\-y=n.  (1) 

2V^  =  V240.  (2) 

The  solution  of  (1)  and  (2)  gives  the  values  of  x  and  y.  The 
most  convenient  method  for  solving  consists  in  subtracting  the 
square  of  (2)  from  the  square  of  (1),  thus  obtaining  {x  —  yf. 

Squaring  (1),  ay^  +  2  xy -{- y' =  289.  (3) 

Squaring  (2),  4:xy  =  240.  (4) 

(3)-(4),  x'-2xy  +  y'  =  A9.  (5) 

Extracting  square  roots,  a;  —  y  =  7.  (6) 


RADICALS  259 

But  x-\-y  =  17.  (1) 

Therefore  ic  =  12  and  2/  =  5. 

Hence  VlT+VMo  =  Vl2  +  V5. 

Vl7  +  V24()  =  ±(2V3+V5). 
If  in  (6)  we  select  —  7  as  the  square  root  of  49,  the  values 
of  X  and  y  are  interchanged,  and  the  final  answer  is  the  same 
as  before. 

Note.  The  preceding  example  gives  a  rational  x  and  y,  because 
172  _  240  is  a  perfect  square.  In  general  the  method  can  be  used  for 
the  finding  of  ^a  +  Vfe  only  if  ai^  —  &  is  a  perfect  square. 


Ex.   Find  V2  a  -  V4  a'  -  4  h\ 


Let  V  2  a  -  V4  a2  -  4  ft2  =  Vx  -  v^. 

Squaring  both  members,  2  a  —  \/4  cC^  —  4  6-  =  x  —  2Vxy  +  y. 
Then,  by  §  274,  re  +  y  =  2  a.  (1) 

2  Vxy  =  V4  a2  _  4  52.  (2) 

Squaring  (1),  a;2  +  2  x?/  +  ?/2  =  4  a^.  (3) 

Squaring  (2),  4  a;?/  =  4  a^  -  4  62.  (4) 

(3)-(4) ,  x^-2xy-\-  y^  =  4  b^. 

Extracting  the  square  root,  x  —  y  =  2b. 

But  x  +  y  =  2a' 

Therefore  x  =  a  +  b,  y  =  a  —  b, 

and  ^2  a-  V4  a^  -  4  62  -  ^  (  Va  +  6  -  Va  -  b). 

EXERCISE  105 

Extract  the  square  roots  of  the  following  binomials : 

1.  7  +  2ViO.  5.    7+V40. 

2.  3  +  2V2,  6.    19-4Vi2. 

3.  6  +  V32.  7.    8-V55. 

4.  7  +  V24.  8.   46-3V20. 


260  ELEMENTARY  ALGEBHA 

9.  9  +  4V5.  14.   33-10V8. 

10.  83-f-12V35.  '        15.    m  +  w  +  2Vmii. 

11.  52 +  14 Vs.  16.    2a-^h-2Va--\-ab. 


12.  69  +  16V5.  17.    a2  +  2a  +  2Va3-l. 

13.  7-V33.  18.    2  4-2a2-V4(l  +  a2  +  a^) 

276.    To  find  the  square  root  of  a  binomial  square  by  inspection. 
According  to  §  65, 

( V5  4- V3)2  =  5  4- 2  V5T3  +  3. 
=  8  +  2Vi5. 


If  on  the  other  hand  we  had  to  find  V 8  +  2 Vl5,  the 
problem  would  be  quite  simple,  if  presented  in  the  form 
V5  _}-2V3  -5+3.  To  reduce  it  to  this  form,  we  must  find 
two  numbers  whose  sum  is  8  and  whose  product  is  15,  viz. 
5  and  3. 


Ex.  1.     Find  'V12-f2V20. 


Find  two  numbers  whose  sum  is  12  and  whose  product  is  20.     These 
numbers  are  10  and  2; 


V12  +  2\/20  =  V 10  +  2\/l0  X  2  +  2. 
=  ViO  +  V2. 


Ex.2.     Eind  Vll_6V2. 

Write  the  binomial  so  that  the  coefficient  of  the  irrational  terra  is  2. 


^^11  -^  6  V2  =  ^11  -  2  VI8. 

Find  two  numbers  whose  sum  is  11,  and  whose  product  is  18.     The 
numbers  are  9  and  2. 

Hence  ^11 -6V2  =^9  -  2\/2T9  +  2. 

=  \/9  -  v/2. 
=  3-v'2. 


RADICALS  261 

•  .. 

EXERCISE   106 

Find  by  inspection  the  square  roots  of  the  following  expres- 
sions : 

1.  3-2V2.  5.  7-2VIO.  8.  11-4V6. 

2.  5-2V6.  6.  7-Vi8.  9.  6+V32. 

3.  4-2V3.  7.  11-4V7.  10.  30  +  12V6. 

4.  6  +  2V5. 

Find  the  fourth  roots  of : 

11.   I7-I2V2.  12.   28-f-16V3. 

RADICAL  EQUATIONS 

277.  A  radical  equation  is  an  equation  involving  an  irrational 

root  of  an  unknown  number. 

Vx  =  5,    v^x  +  3  =  7,   (2  X  —  x2)5  =  1,  are  radical  equations. 

278.  Radical  equations  are  rationalized,  i.e.  they  are  trans- 
formed into  rational  equations,  by  raising  both  members  to 
equal  powers. 

Before  performing  the  involution,  it  is  necessary  in  most 
examples  to  simplify  the  equation  as  much  as  possible,  and  to 
transpose  the  terms  so  that  one  radical  stands  alone  in  one 
member. 

If  all  radicals  do  not  disappear  through  the  first  involution, 
the  process  must  be  repeated. 


Ex.1.    Solve  Va^+12-a;  =  2. 


Transposing  x,  VaT^  +  12  =  x  4-  2. 

Squaring  both  members,  x^  +  12  =  x^  +  4  x  +  4. 

Transposing  and  uniting,  —  4  x  =  —  8. 

Dividing  by  —  4,  x  =  2. 

Check.    The  value  x  =  2  reduces  each  member  to  2. 


262  ELEMENTARY  ALGEBRA 


Ex.  2.    Solve  6 Vic  +  5-3  =  Wx  +  5  +  17. 


Transposing  and  uniting,  2  Vx  +  5  =  20. 


Dividing  by  2,  Vx  +  5  =  10. 

Squaring  both  members,  x  +  b  =  100. 

Transposing,  x  =  95. 

Check.     6V95  +  5  -  3  =  57  ;  4V95  +  5  +  17  =  57. 

Ex.  3.    Solve  Va;  +  5  +  V^  -  V4  a;  +  9  =  0. 

Transpose  V4  x  -f  9,  Vx  +  5  -|-  Vx  =  \/4x  +  9. 

Squaring  botli  members,  x  +  5  +  2Vx"-^  +  5x  +  x  =  4  x  +  9. 
Transposing  and  uniting,  2a/x'^  +  5x  =  2 x  +  4. 

Dividing  by  2,  Vx^  +  5  x  =  x  +  2. 

Squaring  both  members,  x^  +  5  x  =  x^  +  4  x+  4. 

Transposing  and  uniting,  x  =  4. 

Check.     V4+^  +  Vi  -  VlO  +  9  =  3  +  2-5  =  0. 


3 


Ex.4.    Solve  Vl4-a;  +  Vll-a7=     ____ 
Clearing  of  fractions,  \/l54  —  25  x  +  x'^  +  11  -  x  =  3. 


Transposing,  vl54  —  25  x  +  x-^  =  x  —  8. 

Squaring,  154  -  25  x  +  x2  =  x^  -  16  x  +  64. 

Transposing  and  uniting,  —  9  x  =  —  90. 

Dividing  by  —  9,  x  =  10. 

3 


Check.     V 14  -  10  +  Vl  1  -  10  =  3 ; 


Vll  -  10 


EXERCISE  107 
Solve  the  following  equations : 

1.  Vx  =  a.  5.    5-3V2a;-l  =  2. 

2.  ViC  +  5  =  7.  6.    ■Vx  —  a  —  b  =  c. 

3.  5  +  v^=8.  7.   13-V4a^  +  7a;-8=2aj. 

4.  5  +  3Va5=7.  8.  V(a;-5)(4a;+4)  +  6=2». 


RADICALS 


263 


9.   4a;-V(2a;  +  5)(8a;- 7)+7  =  6. 


10.  ■Vixf-\-12x  =  x-^2. 

11.  ^  +  2  =  3. 


12.    Vx-\-7  —Vx—5  =  2. 


13.    V3a;  +  10  =  7-V3a;-ll. 


14.    V7¥+l  =  V7a;  +  18-1. 


15.  V4a.-4-l4-V4a;  +  25  =  12. 

16.  Vaj  —  3  —  1  =  Vx  —  10. 


17.    V5x  +  6  +  2  =  V5»  +  34. 


18.  V9a;-ll  +  3  =  V9a;  +  28. 

19.  Vo;  +  60  =  2  Va;  4-  5  4-  V^. 


20.    Vaj  +  9-Vx  +  2=  V4a;  — 27. 


21.    Vl2a;-ll-V3a;  +  l=V3aj-6. 


22.    V9a;-2  =  V25a;-ll-V4a;-3. 


27. 


28. 


29. 


23.  V9a;-5-2V4a;-15-Va;-5  =  0. 

24.  ^6  +  V4  +  Va;  +  2  =  3. 

25.  (2V^  +  3)(2V^- 3)  =  7. 

26.  (3V^+2)(3V^-2)  =  5. 
2V^  +  1  2V^  +  3 


3V^-2     3Va;-5 

4V^-  — 7_Vx  — 7 
5V»-6  Va;-6 
11-V25^^  V^  +  2 

27  —  5  Vx      ^/x  —  4 
5x-\-C^    _ 


31.    Va;-|-4  = 


x-hl 


30.    V7a;  +  2  = 


Va;-1 

32.    V9¥TlO  =  -5^±^. 

V4a;  +  9 

33.  V2^:=l=  ^("-^)-. 

V2a;-10 

34^     6.T-8V^  +  8  ^o 

4a;-7V»  +  12 


264  ELEMENTARY  ALGEBRA 

4 


35.  Vl9  —  X -h  Vll  —  X 

36.  -Va  —  x-{--\/b  —  x  = 


b 


■Vb  —  x 


37.    Vx-j-2-Vx'^  =  Vx-lS-Vx-25. 


38.    ■Vx-7  —  Vx—10  =  Vx-\-5  —  -Vx  —  2. 
2Vax  —  b_3Vax  —  2b^  3V3x_-f_5  _ 

.'   2^ax  +  b     3Va^  +  46  *    3V3^-5 


41.  V35  +  7V5a;  +  4  =  14. 

42.  (1-V^):(1+3V^)  =  1:4. 

43.  (44-V^):(4-V^)  =  4:l. 

44.  (a  -f-  Va?)  :  (a  —  -y/x)  =  a  -{-  b  :  a  —  b. 

a*  [7  Vx  — 4V2/  =  11- 


45. 

47.    V3^-V2«4-Va;  =  -2V2. 
48. 


6Va;— 5  — 5V2/  +  4=   4. 
,  13  V^^^+  2VF+4  =  60. 


EXTRANEOUS  ROOTS  AND  EQUIVALENT  EQUATIONS 

279.    Solve  the  equation : 

5+V^  =  2  (1) 

Transposing,  s/x  =  —3  (2) 

Squaring  both  members,     x  =  9.  (3) 

But  the  root  a;  =  9  does  not  satisfy  the  equation  (1)  since, 
5  +  V9  =  8. 

*  Consider  the  equation  a  proportion   and   apply    composition    and 
division. 


BADICALS  266 

If  the  value  of  V9  had  not  been  restricted  (§  255)  to  its  posi- 
tive value,  9  would  be  the  root.  On  the  other  hand,  no  positive 
value  of  -y/x  can  satisfy  the  equation,  since  5  plus  a  positive 
number  cannot  equal  2. 

Consequently  equation  (1)  has  no  root,  although  the  student 
should  bear  in  mind  that  this  is  the  consequence  of  the  arbi- 
trary exclusion  of  negative  values  of  roots. 

280.  If  «  =  9  is  not  a  root  of  the  equation  5  +  Vx  =  2,  it 
becomes  necessary  to  ascertain  why  the  solution  produced  this 
value. 

In  solving  an  equation,  we  usually  proceed  as  if  the  given 
equation  were  true  and  we  had  to  prove  the  correctness  of 
each  following  one ;  or  we  prove  that  (3)  is  true  if  (1)  is  true. 
The  real  problem,  however,  is  the  opposite  one,  viz.  to  prove 
equation  (1)  is  true  if  equation  (3)  is  true.  That  is,  we  should 
start  from  equation  (3)  and  prove  successively  (2)  and  (1). 

But  the  members  of  (2)  are  the  square  roots  of  the  members 
of  (3),  and  if  two  quantities  are  equal,  their  square  roots  are 
not  necessarily  equal,  as  shown  by  the  following  illustration: 
(—  3)2  =  (+  3)",  while  —  3  does  not  equal  +3.  Hence  (2)  does 
not  need  to  be  true,  if  (3)  is  true. 

In  general,  squaring  the  two  members  of  an  equation  intro- 
duces a  new  root,  as  can  be  seen  from  the  following  example : 

Let  x  =  a. 

Squaring  both  members,   a^  =  a^. 

The  roots  of  the  second  equation  are  -\-  a  and  —  a,  while  the 
first  one  has  only  one  root,  +  a. 

281.  Equivalent  equations  are  equations  which  have  the  same 

roots ;  as  ic  -|-  4  =  -s/x  and  x  =  Va?  —4. 

282.  A  new  root  which  is  introduced  by  performing  the 
same  operations  on  both  members  of  an  equation  is  called  an 
extraneous  root. 


266  ELEMENTARY  ALGEBRA 

283.  Squaring  both  members  of  an  equation  frequently 
introduces  an  extraneous  root. 

284.  Multiplying  both  members  of  an  equation  by  an  ex- 
pression involving  x  usually  introduces  an  extraneous  root. 

E.g.  a;  — 4  =  0  has  one  root,  a;  =  4. 

Multiplying  both  members  by  x,  we  obtain 

a^-4x  =  0, 

an  equation  which  has  two  roots,  4  and  0. 

285.  The  results  of  a  radical  equation  must  be  substituted  in 
the  given  equation  to  determine  whether  the  7'oots  are  true  roots  or 
extraneous  roots. 

EXERCISE   108 

Solve  the  following  equations,  and  if  the  resulting  roots  are 
extraneous,  change  the  equations  so  as  to  make  the  answer 
true  roots :  

1.  5-Va;  +  4  =  6. 

2.  Vic  +  5  —  V^  =  V4a7  +  9. 


3.  Va;+9-Va;  +  2  =  V4a;-27. 

4.  Va;  +  T  +  Vic  — 5  =  2. 


5.  Va;  +  3  —  Vi»  -  2  =  5. 

6.  V5  +  a;  +  Va;  — 2  =  7. 


CHAPTER  XVI 

THE    FACTOR   THEOREM 

286.  If  a^  —  3  a;-  +  4  a;  -}-  8  is  divided  hj  x  —  2  and  there  is  a 
remainder  (which  does  not  contain  x),  then 

flj^  —  3a^+4£c+8=(a;  —  2)  x  Quotient  +  Eemainder. 

Or,  substituting  Q  and  B  respectively  for  "Quotient"  and 
"Remainder,"  and  transposing. 

As  E  does  not  contain  x,  we  could,  if  Q  was  known,  assign 
to  x  any  value  whatsoever  and  would  always  obtain  the  same 
answer  for  B. 

If,  however,  we  make  x  =  2,  then  (x  —  2)Q=0,  no  matter  what 
the  value  of  Q.  Hence,  even  if  Q  is  unknown,  we  can  find  the 
value  of  E  by  making  x  =  2. 

22  =  23-3.22  4-4.2  +  8-0  =  12. 

Ex.  1.  Without  actual  division,  find  the  remainder  obtained 
by  dividing  3  a;''  +  2  a;  —  5  by  a;  —  3. 

Let  a;  =  3, 

then  2?  =  3.  81  +  2. 3-5-0  =  244. 

Ex.  2.  Without  actual  division,  find  tlie  remainder  when 
ax*  -{- bx^ -{- coiy^ -\- dx -\- e  is  divided  by  x  —  m. 

B  =  ax:^  -^  bx^  +  cx^ -\- dx  +  e  -  (x  -  m)  Q. 
Let  X  =  m, 

then  B  =  am*  +  bm^  +  cm^  +  dm  +  e. 

267 


268  ELEMENTARY  ALGEBRA 

287.  The  Remainder  Theorem.  If  an  integral  rational  expres- 
sion involving  x  is  divided  by  x  —  m,  the  remainder  is  obtained 
by  substituting  in  the  given  expression  m  in  place  ofx. 

E.g.     The  remainder  of  the  division 

(4  a:5  _  4  a;  +  11)  ^  (x  +  3)  is  4  (-  3)5  _  4  (-  3)  +  11  =  -  949. 
The  remainder  obtained  by  dividing 

(x  4-  4)*  -  (x  +  2)(x  -1)  +  7  by  X  -  1  is  5*  -  3  .  0  +  7  =  632o 

EXERCISE  109 

Without  actual  division,  find  the  remainder  obtained  by 
dividing : 

1.  a^-f3x2-3x  +  2by  aj-4. 

2.  x^'^-la?-\-4.x^-2x-^hj  x-1, 

3.  a«  +  2a2-4a  +  lbja-3. 

4.  a*4-5a2-|_2a4-l  by  a-l-2. 

5.  (a:  +  l)'-3(a;-2)(x  +  2)  +  (i«  +  l)2by  aj-2. 

6.  0^  —  4,  x-m  -f  4  xrn^  +  m^hy  x  —  m. 

7.  x^  ~2xhi-^4:Xn^ —  n^hy  x  +  2n. 

8.  (x-^A.Y-\-{x  +  Sf-{-(x  +  2Yhjx  +  l, 

9.  (a-l)(a-3)-4[(a-3)(a-4)-3]  by  a-a 

10.  ^-\-Why  x  —  b. 

11.  a^-{-b^hj  a  +  b. 

12.  a^  +  b'^hj  a  +  b. 

288.  If  the  remainder  is  zero,  the  divisor  is  a  factor  of 
the  dividend. 

The  Factor  Theorem.  If  a  rational  integral  expression  involv- 
ing X  becomes  zero  when  m  is  written  in  place  of  x,  x  —  m  is  a 
factor  of  the  expression. 

E.g.  if  x8  —  3^2  —  2  a;  —  8  is  divided  by  x  —  4,  the  remainder  equals 
43  _  3  .  42  -  2  .  4  -  8  =  0,  hence  (x  -  4)  is  a  factor  of  x^  -  3  x-  -  2  x  -8. 


THE  FACTOR   THEOREM  269 

289.  A  rational  integral  expression,  aa;"  + ftaj""^  •••  ea;-|-/,  is 
divisible  hy  x  —  m  only,  if  m  is  a  factor  of/,     (§  131.) 

Hence,  to  factor  the  expression,  we  substitute  for  m  exact 
divisors  of  /,  and  determine  by  means  of  the  factor  theorem 
whether  «  —  m  is  a  factor  or  not. 

Ex.  1.   Factor  a^-7x^  +  7x-\- 15. 

The  exact  divisors  of  15  are  +1,  —  1,  +3,  —  3,  +5,  —  5,  +15,  — 15. 

Let  X  =  1,  then  x^  —  Tx^  +  Tx  +  lS  does  not  vanish. 
Let  X  =  -1,  then  x^  -  7  x'^  +  7  x  +  W  =  0. 
Therefore  x—{-  1),  orx  +  1,  is  a  factor. 
By  dividing  by  x  +  1 ,  we  obtain 

x3  -  7  x2  +  7  X  +  15  =  (x  +  l)(x2  -  8  X  +  15). 
=:(x  +  l)(x-3)(x-5). 

Ex.  2.    Factor  sc^^bx^-a^x-  a^b. 

The  exact  divisors  of  a^b  are  —  a,  +  a,  —  6,  +  6,  —  ah,  etc. 
The  substitution  x  =  a  makes  the  expression  vanish. 
Dividing  by  x  -  a,  we  have 

x3  +  bx^  -  a^x  -  a^b  =  (x  -  a)  (x^  +  6x  +  ax  +  ab), 
=  (x  —  a)  (x +  «)(«  + &). 

EXERCISE  110 
Without  actual  division,  show  that : 

1.  10  ic^  —  4  X*  —  13  a;2  +  7  is  divisible  by  a;  —  1. 

2.  a*  —  4  a^  —  7  a  —  24  is  divisible  by  a  —  3. 

3.  a*  +  a^—  ab^  —  b^  is  divisible  by  a  —  6. 

4.  a;^  +  3  a^  —  4  a;  — 12  is  divisible  by  a?  —  2  and  x-\-2. 

5.  (a:  +  1)^  (»- 2) -4  (a; -1) (a; -3) +  4  is  divisible  by  a; -1. 

6.  6a;[4(a;4-l)(a;  +  2)-47]-a;«  — a2  +  3a;-6    is    divisible 

by  x-2. 

7.  6ar'-3a;^-5a3  +  5ar*-2a;-3  is  divisible  by  a;  +  l. 


270  ELEMENTARY  ALGEBRA 

Resolve  into  factors : 

8.  a^  — 2a-4.     ,  12.    cc^ -7  x -{-6. 

9.  a^-\.2x-3.  13.    n^-7n-6. 

10.  Sa^-Sx  +  5.  14.   _p3  +  7p2  +  i4p_p8. 

11.  a'^  +  6a2  +  lla  +  6.  15.    m^-igm  +  SO. 

16.  4a^-13a;  +  6. 

17.  m'^-3m^n-5mn^  +  lSn\ 

18.  2a^-5x2-13a;  +  30. 

19.  a;^  — 0^  — Taj^  +  aj  +  e. 

20.  6af  +  7  x^-x-2. 

21.  a;3-(3a-6y  +  (2a2-3a%  +  2a2&. 

22.  ar'  —  (a  +  5  +  c)x^  +  (a6  +  &c  +  ca)a;  —  ahc. 

23.  aj3  -(3  a  +  2  6)a;2 4-(6  ab-{-2  a')x-A  a^b. 

24.  Find  the  H.  C.  F.  of  3  a^-i-5  a^-a-^2  and  d'-^-a^-a^-^. 

25.  Find  the  H.  C.  F.  of  9  a^  +  18  aj^  -  a;  - 10  and  3  aj^  + 13  ar' 
+  2  a;  -  8. 

26.  Find    the    H.C.F.    of    aj^-oj^- 5  aj-3    and    a^-4a^ 
-11  a;- 6.  

290.    If  n  is  a  positive  integer,  it  follows  from  the  Factor 
Theorem  that 

1.  a;"  —  2/"  is  always  divisible  by  x  —  y. 

For  substituting  y  for  x,  a;"  —  ?/"  =  ?/'•  —  z/"  =  0. 

2.  a;**  -\- 1/"  IS  divisible  by  x  +  y,  ifn  is  odd. 

For  (-  yy  +  y''  =  0,  if  7i  is  odd. 

By  actual  division  we  obtain  the  other  factors,  and  have  for 
any  positive  integral  value  of  n, 

*  The  symbol  •••  means  "and  so  forth  to.'* 


THE  FACTOR   THEOREM  271 

If  n  is  odd, 

a?**  4-  2/"  =  (»  +  2/)(«''~V—  ^""^2/  +  aj^-y h  2/""^). 

,   e.gr.  jkS  _  ^5  _  (a;  _  y)  (a;4  +  a;3y  +  a;2y2  +  xy^  +  y*). 
a;^  +  2/^  =  (a:  +  y) (x*  -  a^^y  +  xV  -  xy^  +  ?/4). 

291.  It  can  readily  be  seen  that  a;** +  2/**  is  not  divisible  by 
either  x-\-y  or  x  —  y,  if  w  is  even,  and  that  a;"  —  ?/"  cannot  be 
divided  hj  x  +  y,\i  n  is  odd. 

EXERCISE  111 

State  whether  the  following  expressions  are  prime  or  not, 
and  factor  whenever  possible. 

1.  a^-1.  4.    m^-{-n\  7.    l  +  aJ.  10.    a*  —  b^^. 

2.  a^  +  1.  5.    m^  —  n\  8.    1  — a^  11.    6^  — a^. 

3.  a^  +  h\  6.    a^-l.  9.    a^^-fe^^        12.    a^^-ft*. 

[For  additional  examples  of  this  type,  see  Appendix  II.] 


CHAPTER   XVII* 

GRAPHIC   REPRESENTATION   OF   FUNCTIONS 
AND   EQUATIONS 

REPRESENTATION    OF    FUNCTIONS  OF  ONE  VARIABLE 

292.  An  expression  involving  one  or  several  letters  is  called 
a  function  of  these  letters. 

a:^  —  X  +  7  is  a  function  of  x. 

2  a:?/  —  ?/2  +  3  ?/3  is  a  function  of  x  and  y. 

293.  If  the  value  of  a  quantity  changes,  the  value  of  a 
function  of  this  quantity  will  change,  e.g.  if  x  assumes  succes- 
sively the  values  1,  2,  3,  4,  x^  —  cc  +  7  will  respectively  assume 
the  values  7,  9,  13,  19.  If  x  increases  gradually  from  1  to  2, 
0?  —  x-\-l  will  change  gradually  from  7  to  9. 

294.  A  variable  is  a  quantity  whose  value  changes  in  the 
same  discussion, 

295.  A  constant  is  a  quantity  whose  value  doe^  not  change  in 
the  same  discussion. 

In  the  example  of  the  preceding  article,  x  is  supposed  to  change,  hence 
it  is  a  variable,  while  7  is  a  constant. 

296.  A  convenient  method  for  the  representation  of  the 
various  values  of  a  function  of  a  letter,  when  this  letter 
changes,  is  the  method  of  representing  these  values  graphi- 
cally ;  that  is,  by  a  diagram.  This  method  is  frequently  iised  to 
represent  in  a  concise  manner  a  great  many  data  referring  to 
facts  taken  from  physics,  chemistry,  technology,  economics,  etc. 

♦  This  chapter  may  be  omitted  on  a  first  reading. 
272 


GRAPHIC  REPRESENTATIONS 


273 


.11^ 


297.  To  give  first  a  diagram  of  one  of  these  applications,  let 
us  suppose  that  we  have  measured  the  temperatures  at  all 
hours  from  12  m.  to'  11  p.m.  on  a  certain  day,  and  that  we  wish 
to  represent  the  results  graphically.  Draw  a  line  OX,  and 
lay  off  on  it  equal  parts,  representing  the  hours.  Then  0  rep- 
resents 12  o'clock ;  4,  four  o'clock,  etc.  At  each  point  draw  a 
perpendicular  line,  and  make  it 
equal  to  the  temperature  of  the 
corresponding  hour  (any  conven- 
ient length  being  taken  as  a  unit). 

By  inspection  of  the  diagram 
it  can  be  seen  that  the  tempera- 
ture at  12  o'clock  was  3°,  at  1  p.m. 
5°,  at  2  P.M.  61°,  etc. 

But  it  is  possible  to  represent 
in  the  diagram  the  temperatures  of  any  particular  time  between 
12  M.  and  11  p.m.  ;  thus  the  perpendicular  AB  indicates  that 
the  temperature  at  3.30  was  7°.  We  may  also  omit  the  per- 
pendicular and  simply  draw  its 'end  point;  as  point  C.  By 
measuring  the  distance  of  C  from  OX,  we  find  that  the  tem- 
perature at  11.20  was  —3°. 


1     2    3^4-    5    6    7    8    9     I 


L^f^X 


298.  If  we  would  represent  the  temperatures  of  every  mo- 
ment between  12  and  11.20,  we  would  obtain  an  uninterrupted 
sequence  of  points,  or  a 
curved  line,  as  shown  in  the 
next  diagram.  This  curve  is 
said  to  be  a  graphical  represen- 
tation or  a  graph  of  the  tem- 
peratures from  12  to  11.20. 

To  find  from  the  graph  the 
temperatures  at  any  hour, 
e.g.  2.30,  take  a  point  A,  2\ 
units  from  0,  and  measure 
the  length  of  the  perpendicular  at  A  (not  drawn  in  the  diagram). 


Y 

c° 

/ 

/^ 

, 

N 

/ 

\ 

'^ 

\ 

t 

N 

\ 

y 

\ 

/ 

\ 

^ 

\ 

V 

X 

0 

1 

)    t 

i     ( 

) 

N 

01 

\9.}i. 

V 

M 

L. 

274 


ELEMENTARY  ALGEBRA 


^-  ^^ 

l'^^  ^ 

J         \ 

t          \ 

I                   ^v 

A-              \ 

tf               % 

\     ^ 

0       1       i34     56     7     89\|0llP-'»f. 

V 

X^ 

EXERCISE   112 

From  the  diagram  find  approximate  answers  to  the  following 
questions : 

1.  Determine  the  tem- 
perature at: 

(a)  5  P.M.     (b)  1.30  P.M. 
(c)  5.45  P.M.     (d)  11.45  A.M. 

2.  At  what  hour  or  hours 
was  the  temperature  (a)  6°, 
(b)5°,(c)V,{d)-V,(e)0''? 

3.  At  what  hour  was 
the    temperature    highest  ? 

4.  What  was  the  highest  temperature  ? 

5.  During  what  hours  was  the  temperature  above  5°  ? 

6.  During  what  hours  was  the  temperature  between  3°  and 
4°? 

7.  During  what  hours  was  the  temperature  above  0°  ? 

8.  During  what  hours  was  the  temperature  below  0°  ? 

9.  How  much  higher  was  the  temperature  at  4  than  at 

8    P.M.? 

10.  At  what  hour  was  the  temperature  the  same  as  at  1  p.m.  ? 

11.  During  what  hours  did  the  temperature  increase  ? 

12.  During  what  hours  did  the  temperature  decrease  ? 

13.  Between  which  two  successive  hours  did  the  temperature 
change  least  ? 

14.  Between  which  two  successive  hours  did  the  temperature 
increase  most  rapidly  ? 

299.  Graph  of  a  function.  To  represent  the  various  values 
which  an  algebraic  function,  e.g.  3  -{-  ^x  —  \x^,  assumes,  when 
X  changes  from  —3  to  +9,  let  us  substitute  in  3  +  f «  — Ja;^, 
successively  the  values  —  3,  —  2,  —  1,  0,  1,  etc. 


GBA  PHIC  R  EP  BE  SENT  A  TIONS 


275 


If  a;  respectively  =- 3,    -2,-1,    0,1,    2,3,    4,5,    6,7,        8,      9, 
then  3  +  f  x  -  ix2  =  -  3|,  -  1,      l^,  3,  4^,  5,  5^,  5,  4^,  3,  1|,  -  1,  -3|. 

Selecting  any  convenient   length   as  a  unit,  we  lay  off  on 
a   line    OX   from    0    the   values    of    x,  and    at    each    point 

erect  a  perpendicular    equal 
to  the  corresponding  value  of 

To  find  e.g.  the  point  repre- 
senting that, 

for  a;  =  2,3  +  fa;-ia^  =  5, 

we  lay  off  OA  =  2,  and  at  A 
draw  a  perpendicular  ^5=5. 
Similarly  for  all  other  points, 
as  C,  D,  etc.  Since  the  posi- 
tive values  of  x  are  laid  off  from  0  toward  X,  negative  values  of  x 
must  be  laid  off  from  0  in  the  opposite  direction  ;  as  0F=  —  2. 
Similarly  negative  values  of  3  +  f  a? 
a  direction  opposite  to  the  positive  values,  as  FG  = 


■ 

Y 

B  C  p   1 

I 

- 

1 

1    F 

- 

-3   - 

2 

A 

8     < 

X 

1     - 

1 

0 

1   : 

; 

' 

■ 

J    ( 

) 

G 

1 

\oc^  must  be  laid  off  in 


1,  or 


HI= 


3}. 


300.  The  lines  OA  and  AB  are  called  the  coordinates  of 

point  B,  OA  is  the  abscissa,  AB  the  ordinate  of  point  B.     The 

abscissa  of  I  is  OH,  its  ordinate  HI. 

« 

301.  The  line  OX  is  called  the  axis  of  abscissas  or  jr-axis.  The 
perpendicular  OY,  erected  at  0,  is  the  axis  of  ordinates  or^-axis. 
The  point  0  is  called  the  origin.  Abscissas  measured  to  the 
right  of  the  origin,  and  ordinates  above  the  ic-axis,  are  considered 
positive. 

302.  The  point  whose  abscissa  is  x  and  whose  ordinate  is  y 
is  often  represented  by  {x,  y).  Thus  (2,  5)  represents  the  point 
B,  (-2,  -1)  is  the  point  G. 

The  point  (a;,  y)  can  also  be  obtained  by  first  laying  off  y  on  the  y-axis, 
and  then  drawing  x  perpendicular  to  oy  at  the  extremity  of  y. 


276 


ELEMENTARY  ALGEBRA 


303.  If  we  should  rep- 
resent the  values  of  the 
function  3  -f  f  a?  —  i  ic^  for 
all  values  of  x  from 
x=  —  3  to  x=  -^9)  and 
should  omit  the  perpendic- 
ulars, we  should  obtain  a 
continuous  succession  of 
points,  or  a  curved  line 
as  shown  in  the  diagram. 

This  curve  is  called  the 
graph  of  the  function. 

Note.  It  is  convenient  to  use  for  such  drawings  coordinate  paper, 
i.e.  paper  divided  into  squares,  as  sliown  in  the  diagram.  An  ordinary 
ruled  sheet  can  also  be  used  to  advantage. 


Y 

^ ^ 

T    ^        ^v 

/       _.^ 

/             \ 

/                   \ 

7-  °             I     I 

X^-  -3  -2/  -1            1     2     3     4     5     6     7    \8     9 

Tt^                ^ 

lA                             A 

'  r                      ^^ 

EXERCISE   113 

Find  from  the  diagram  of  the  preceding  article  approximate 
answers  to  Exs.  1-19. 

1.  Determine   the  value  of   the  function,  S  +  ^x  —  {x^,  if 
(a)  a;  =-21,    (/>)  a;  =  5,   (c)  x  =  6^,    (d)  x  =  8|. 

2.  What  value  or  values  of  x  will  make  3-f-f  r»  — Ja^  equal 
to(«)3,  (6)4,(c)-2,  (d)  1,  (e)0?^ 

3.  What   is   the    greatest    value    which    S-{-^x  —  \x^  can 
assume  between  a;  =  —  3  and  x  =  -\-9? 

4.  What  value  of  x  produces  the  greatest  value  of  3  -h  |  cc 

5.  What  values   of  x  produce   positive   values   of   3-f-fa; 

6.  What  values  of  x  produce  negative  values  of  the  same 
function  ? 

7.  Between  what  values  must  x  lie  to  produce  a  function 
3  _^  3  a;  _  1  a^  greater  than  -f  3  ? 


GRAPHIC  REPRESENTATIONS  277 

8.  What  values  of  x  produce  values  of  the  same  function 
smaller  than  —  1  ? 

9.  What  values  of  x  make  the  same  function  equal  to  zero  ? 

10.  What  then  are  the  roots  of  the  equations  3  +  f  .t  — Ja;^ 
=  0? 

11.  Find  two  roots  of  the  equation  3-f|a;  —  Ja^  =  2. 

12.  Find  two  roots  of  the  equation  3+f£c  —  ^a^  =  3. 

13.  Find  two  roots  of  the  equation  3  +  fa;  —  Jic^  =  4. 

14.  Has  the  equation  3-{-^x  —  ^x^=6  any  real  roots  ? 

15.  Find   the  value  of   m  for  which  the  equation  3  4- fa; 
—  :^x^  =  m  has  only  one  root. 

16.  How  much   smaller  is  the  value  of  the  function  for 
x  =  6  than  for  a;  =  3  ? 

17.  Which  other  value  of  x  produces  the  same  value  of  the 
function  as  x  =  6  ? 

18.  If  x  increases  from  —3  to  4-2,  does  the  function  also 
increase  ? 

19.  Up  to  what  value  of  x  will  the  function  increase  when 
X  increases  ? 

20.  Locate  the  points  (2,  5),  (3,  6),  (3,  -  2),  (0,  5),  (-2,  5), 
(-5,  -4),  (-31  0),   (0,  -1),  (-1,0). 

21.  Locate  the  points  (-2,  0),  (0,  -2),  (- 1   -i). 

22.  Locate  the  points  (-  3,  3),  (-  2,  3),  (- 1,  3),  (0,  3),  (1,  3). 

23.  Where  do  all  points  lie  whose  ordinate  is  3  ? 

24.  Locate  the  points  representing  the  values  of  fa;  + 1,  if 
a;--3,  -2,  -1,0,1,2,3. 

25 .  Construct  the  graph  of  the  function  |  a;  -f  1,  from  a?  =  —  3, 
to  a;  =  +  3. 

26.  From  the  graph  constructed  in  Ex.  25  find  approxi- 
mately : 

(a)  the  value  of  fa;  +  1  if  a;  =  —  2 J,  —^,  1^. 
(6)  the  value  of  a;,  if  f  a?  + 1  =  -  i,  2,  2^. 


278  ELEMENTARY  ALGEBRA 

(c)  the  values  of  x  whicli  make  f  a;  -f  1  positive. 

(d)  the  values  of  x  which  make  |  a;  +  1  negative. 

(e)  the  value  of  x  that  makes  |  a?  +  1  =  0. 
(/)  the  root  of  the  equation  fa?  +  1  =  0. 
(g)  the  root  of  the  equation  |  a;  + 1  =  2. 

27.  Construct  the  graph  of  x~,  from  x  —  —  5,  to  a;  =  +  5,  and 
from  the  diagram  determine  the  following  values:  (— l^)-, 
(3\y,  (2.2)^  V7,  Vi2,  VO.  (Make  the  scale  unit  of  the  x 
equal  to  10  times  the  scale  unit  of  the  a^.) 

GRAPHIC   SOLUTION    OF    EQUATIONS    INVOLVING   ONE 
UNKNOWN   QUANTITY 

304.  A  rational  integral  equation  which  contains  the  nth. 
power  of  the  unknown  quantity,  but  no  higher  power,  is  called 
an  equation  of  the  nth  degree. 

ic^—  2 oj  —  4  is  an  equation  of  the  third  degree. 

305.  The  roots  of  equations  of  the  first  and  higher  degrees  can 
be  found  approximately  by  the  graphical  method. 

Ex.     Find  graphically  the  roots  of  the  equation  ^cc'^+r^— 2  =  0. 

To  obtain  the  values  of  the  function  for  the  various  values 
of  X,  the  following  arrangement  may  be  found  convenient : 

(Compute  each  column  before  commencing  the  next,  and  see  table  on 
p.  288.) 


X 

x^ 

a;3 

ix^ 

ix^+x^ 

^  »-3  +  X2  -  2 

-5 

25 

-125 

-25 

0 

-    2 

-4 

16 

-    64 

-12.8 

3.2 

1.2 

-3 

9 

-    27 

-    5.4 

3.6 

1.6 

-2 

4 

-      8 

-    1.6 

2.4 

.4 

-1 

1 

-      1 

—      .2 

.8 

-    1.2 

0 

0 

0 

0 

0 

-    2 

1 

1 

1 

.2 

1.2 

-      .8 

2 

4 

8 

1.6 

5.6 

3.6 

3 

9 

27 

5.4 

14.4 

12.4 

GRAPHIC  REPRESENTATIONS 


279 


Locating  the  points  (—5,  —2),  (—4,  1.2),  etc.,  and  joining, 
produces  the  graph  ABC.  Since  ABC  intersects  the  avaxis  at 
three  points,  F,  P',  and  F",  three  values  of  x  make  the  function 
zero.  Hence  there  are  three  roots  which  are  found,  by  meas- 
urement of  OF"  J  OP  J  and  OP,  to  be  approximately  —  4.5, 
— 1.7,  and  1.25. 


To  find  a  more  exact  answer  for  one  of  these  roots,  e.g.  OP,  we  draw 
the  portion  of  the  diagram  which  contains  P  on  a  larger  scale. 

If  X  =  1.25,  the  function  equals  —  .0469,  i.e.  it  is  negative.  Hence  it 
appears  from  the  diagram  that  the  root  must  be  larger,  and  we  substitute 
jc  =  1.26,  which  produces  ^x^  +  a;'^  —  2  =  —  .0123.  This  again  being  a 
negative  value,  we  substitute  x  =  1.27,  which  produces  the  positive  value, 
.0226.     The  root,  therefore,  must  lie  between  1.26  and  1.27. 

Making  the  side  of  each  small  square  (diagram  II)  equal  to  .001,  we 
locate  the  points  (1.26,  -  .0123)  and  (1.27,  .0226),  i.e.  B'  and  C".  Since 
in  nearly  all  cases  small  portions  of  the  curve  are  almost  straight  lines, 
■we  join  the  two  points  by  a  straight  line  B'C,  which  intersects  the  x-axis 
in  P. 

The  measurement  of  OP  gives  the  root 

X  =  1.2637, 


280 


ELEMENTARY  ALGEBRA 


an  answer  whose  last  place  is  not  quite  reliable.    Restricting  the  answer 
to  three  places,  we  obtain  the  root 

X  =  1.264. 

If  a  greater  degree  of  accuracy  is  required,  a  third  drawing  on  a  still 
greater  scale  must  be  constructed. 

EXERCISE  114 

1.   From  the  diagram  of  the  preceding  example  find  approxi- 
mate answers  to  the  following  questions : 

(a)  What  is  the  value  oi  ^a^ -i-x" -2  ii  x  =  -1.2 ?    if  a;  = 
-3.4?     ifa;  =  1.5? 


run 


H- 


02- 


^2T 


\m 


n 


(6)  What  values  of  x  will  make  ^oc^-\-x^-2  equal  to  1? 
equal  to  1.4  ?     equal  to  —  2  ? 

(c)  What  is  the  greatest  value  which  ^x^-{-o^  —  2  can  assume 
for  negative  values  of  a?  ? 

(d)  What  negative  value  of  x  produces  the  greatest  value  of 
the  function  ?  * 


GRAPHIC  REPRESENTATIONS  281 

(e)  What  value  of  x  between  —  3  and  2  produces  the  smallest 
value  of  the  function  ? 

(/)  What  is  the  smallest  value  of  the  function  between 
x  =  —  3  and  x  =  -\-S? 

(g)  Between  what  values  must  x  lie  to  produce  (1)  a  positive 
value  of  the  function ;  (2)  a  negative  value  of  the  function  ? 

(h)  Find  the  roots  of  the  equation ; 

(t)  Find  the  roots  of  the  equation : 

^0(^^x^-2  =  1.2. 
(k)  How  many  roots  has  the  equation : 
^  arH  aj2  -  2  =  3  ? 
Find  the  value  of  this  root,  or  these  roots. 
(I)  How  many  roots  has  the  equation : 
^x^  +  x'-2  =  -2? 
(m)  Between  what  values  must  m  lie"  in  order  to  give  the 
equation  ^  ar  -f  a.-^  -  2  =  m,  three  roots  ? 

(n)  In  the  same  equation,  what  values  of  m  produce  two  roots  ? 

(o)  In  the  same  equation,  between  what  limits  must  m  lie  to 
produce  only  one  root  ? 

(p)  If  X  increases,  between  what  values  of  x  does  the  func- 
tion (a)  increase,  (b)  decrease  ? 

2.  Construct  the  graph  of  2  -|-  a;  —  -J-ic^  from  a;  =  —  2  to  a;  =  9^ 
and  from  the  diagram  determine  approximately ; 

(a)  the  value  of  the  function  if  a;  =  2^,  if  a;  =  —  1^. 

(b)  the  value  or  values  of  x  if  the  function  equals  —  1,  +2. 

(c)  the  values  of  x  that  make  the  function  positive. 

(d)  the  values  of  x  that  make  the  function  negative. 

(e)  the  root  or  roots  of  the  equation  obtained  by  making  the 
function  equal  to  zero. 


282  ELEMENTARY  ALGEBRA 

(/)  the  root  or  roots  of  the  equation  obtained  by  making 
the  function  equal  to  1. 

(g)  the  greatest  value  of  the  function  between  the  given 
points. 

(h)  the  value  of  x  that  produces  the  greatest  value  of  the 
function. 

In  each  of  the  Exs.  3-9  construct  the  graph  of  the  function, 
and  from  the  diagram  determine,  wlienever  possible,  the 
answers  to  questions  (a),  (b),  (c),  (d),  (e),  (/),  and  (g),  given 
in  Ex.  2. 

3.  3  —  I  X,  from  x  =  —  2,tox  =  4,. 

4.  X  -\-l,  from  x  —  —  4,  to  x  =  -{-  4t. 

5.  ^a^  —  x—2,  from  x  =  —  3,  to  x  =  -\-5. 

6.  I  x^,  from  x  =  —  6,  to  x  =  S. 

7. -,  from  X  =  1,  to  a?  =  12. 

X     2 

8.    ^jj  x^  —  X  -\- 1,  from  ic  =  —  4,  to  a;  =  4. 


9.    ^25  —  a^,  from  x  =  —  5,tox  =  5. 

306.  It  can  be  proved  that  the  graphs  of  the  functions  of  the 
first  degree  involving  one  unknown  quantity,  are  straight  lines, 
hence  two  points  are  sufficient  for  the  construction  of  these 
graphs.  (This  is  true  whether  the  scales  of  the  abscissas  and 
ordinates  are  equal  or  unequal.) 

10.  Draw  the  graph  of  4  x— 5. 

11.  Draw  the  graph  of  3  —  2  cc. 

12.  Degrees  of  the  Fahrenheit  scale  are  expressed  in  degrees 
of  the  Centigrade  scale  by  the  formula  C  =  |(F  —  32). 

(a)  Draw  the  graph  of  |  (F  -  32),  from  F  =-  -  5,  to  F  =  40. 
(h)  From  the  diagram  find  the  number  of  degrees  of  Centi- 
grade equal  to  -  1°  F.,  9°  F.,  14°  F.,  32°  F. 

(c)  Change  to  Fahrenheit  readings  - 10°  C,  0°  C,  1°  C. 


GRAPHIC  REPRESENTATIONS  283 

13.  The  formula  for  the  distance  traveled  by  a  falling  body 
is  S  =  i  gt\ 

(a)  Kepresent  i  gt^  graphically  from  ^  =  0  to  ^  =  5.  (Assume 
g  =  10  meters,  and  make  the  scale  unit  of  the  t  equal  to  10 
times  the  scale  unit  of  the  ^  gt\) 

(b)  How  far  does  a  body  fall  in  2^  seconds  ? 

(c)  In  how  many  seconds  does  a  body  fall  30  meters  ? 

Solve  graphically : 

14.   4a;  +  3  =  0.  15.    6x-5  =  0. 

Solve  the  following  equations  by  the  graphical  method,  and 
find  the  greatest  or  the  smallest  value  of  the  function: 

(For  the  squares  and  the  cubes  of  numbers,  see  table  on 
p.  288.) 

16.  ii-2_3^_3^()#  24.  a^  +  a;-3  =  0. 

17.  x--x-^f  =  ^.  25.  ic3-4aJ4-l=0. 

18.  ^J^\x^-^-  =  ^.  26.  x?-2x^-^x^h=^^.\ 

19.  ic2_2a;-9  =  0.  27.  a;^-10aj2  +  8  =  0.$ 

20.  a;2+.y.^._ii_0.  28.  a;*-17a:2^_^_^54^0. 

21.  \y?  —  x—^-=^.  gg^  2^ 4- a;  — 4  =  0. 

22.  6it'-~13a;  +  4  =  0.  ^ 

30.    --2^  =  0. 

23.  ^  ar  +  ic  —  4  =  0.  x 

* 

*  A  more  convenient  method  for  solving,  graphically,  equations  of  the 
second  degree  is  given  in  Chapter  XVIII. 

t  To  avoid  large  ordinates,  make  the  scale  unit  of  ordinates  \  of  the 
scale  unit  of  the  abscissas.  The  same  graph  may  be  obtained  by  dividing 
each  term  of  the  equation  by  5  and  using  equal  scale  units  for  ordinates 
and  abscissas. 

X  Make  th'e  scale  unit  of  the  ordinates  -^^  of  the  scale  unit  of  the 
abscissas. 


284 


ELEMENTARY  ALGEBRA 


GRAPHIC   SOLUTION    OF    EQUATIONS   INVOLVING  TWO 
UNKNOWN    QUANTITIES 

307.  In  §  303  the  graph  of  the  function  S^^x-^a^waa 
constructed  and  discussed.  If  3-{-^x~\x^  be  denoted  by  y, 
then  the  ordinate  represents  the  various  values  of  y,  and  the 
diagram  (p.  276)  represents  the  equation, 


y  =  S  +  ix-iaf. 


(1) 


The  coordinates  of  every  point  of  the  curve  satisfy  equation  (1), 
and  every  set  of  real  values  of  x  and  y  satisfying  equation  (1) 
is  represented  by  a  point  in  the  curve. 

308.  The  curve  representing  an  equation  is  called  the  graph 
or  locus  of  the  equation. 

309.  If  an  equation  containing  two  unknown  quantities  can 
be  reduced  to  the  form  y—f(x),  when /(ic)  represents  a  func- 
tion of  X,  then  the  equation  can  be  repre- 
sented graphically. 

Ex.  1.     Represent   graphically  3  a;  — 
2y  =  2. 

Solving  for  y, 


Sx 


Hence,  if  x  equals 
then  y  equals 


1, 

21, 


0,  1,  2,  3 

-ii    •^i   ^1   '->-■ 


-,Y 

■y/ 

X'-2      -1        7]        2       3    X 

v4 — 

-/-  -3 —  ^ — 

_/ Y- 


Locating  the  points  (  —  2,  —4),  (  —  1,  — 2^), 
etc.,  and  drawing  a  line  through  them,  we 
obtain  the  graph  of  the  equation,  which  is  a 
straight  line. 

310.  The  graph  of  an  equation  of  the  first  degree  involving  two 
unknown  quantities  is  always  a  straight  line,  and  hence  it  can  be 
constructed  if  two  points  are  located.     (§  306.) 


GRAPHIC  REPRESENTATIONS 


285 


Ex.  2.     Draw  the  locus  of  4  a;  4-  3  2/  =  12. 

If  a;  =  0,  2/  =  4  ;  if  2/  =  0,  x  =  3. 

Hence,  locate  points  (0,  4)  and  (3,  0),  and  join 
tliem  by  a  straiglit  line  AB.  AB  is  the  required 
graph. 

Note.  Equations  of  the  first  degree  are  called  linear 
equations,  because  their  graphs  are  straight  lines. 


EXERCISE  115 
Draw  the  loci  of  the  following  equations : 

1.  aj+?/  =  4.  5.    a;  +  2/  =  -10.  9.    12 a; +  152^=48. 

2.  a;  — 2 2/ =  4.  6.    2/  =  — 4.  10.   2/— V^  +  2  =  0. 
3-.    2x-^y  =  12.        7.    x  +  y  =  0.  11.    (l.l)^-2/  =  0. 


2/=0. 


8.    2/ 


12.   xy-\-y  —  ii(?  =  0. 


311.   Graphical  solution  of  a  linear  system. 

To  find  the  roots 
of  the  system. 

2a;  +  32/  =  8,     (1) 
a;-22/  =  2.     (2) 

By  the  method  of 
the  preceding  article 
construct  the  graphs 
AB  and  CD  of  (1) 
and  (2)  respectively. 
The  coordinates  of 
every  point  in  AB 
satisfy  the  equation 
(1),  but  only  one  point 
in'^B  also  satisfies 
equation  (2),  viz.  P,  the  point  of  intersection  of  AB  and  CD, 

By  measuring  the   coordinate   of  P,  we  obtain  the  roots, 
a;  =  3. 15, 2/ =  .57. 


1                                             -   -  -      - 

'^Y' 

. 

^^^ 

:?:  ^s           -          _     _       _ 

^s 

"^s.^                       _____ 

"<=;^ 

s  ^ 

_     5:        __ 

.0.           ^S 

■^                  s 

^^                     i" 

^s~                            ^ 

-1-    _                                      __                       "^N.-                                              £' 

X               '                -         ^l    p    ^^ 

:        :                 :     ^^i^^    :: 

i-                                   -^*'s 

t)                                            ^ ^s,              Cl- 

_ _^<: 1^:-> 

^:        :        ::        '^^    ~     :         "i-^^ 

«s^                                    7? 

^^                                      -   3 

,  ..  T,  r                                                                        ... 

It!    ,  -'^                                -    - 

0*-=^              _        _        _I 

_<:                               I  :            :"" 

TT"                 ,  ,,,     _  . 

286 


ELEMENTARY  ALGEBRA 


312.  The  roots  of  two  simultaneous  equations  are  represented 
by  the  coordinates  of  the  point  (or  points)  at  which  their 
graphs  intersect. 

313.  Since  two  straight  lines  which  are  not  coincident  nor 
parallel  have  only  one  point  of  intersection,  simultaneous 
linear  equations  have  only  one  pair  of  roots. 

314.  Equations  of  higher  degree,  however,  can  have  several 
points  of  intersection,  and  hence  several  pairs  of  roots. 

1.    Solve  graphically  the  following  system: 

j      x^  +  f=   25,  (2) 

\dx-2y=-%.  (1) 


Ex. 


Solving  (1)  for  y,  y  =  v  25  —  x'^. 

Therefore,  if  x  equals  —  5,  —  4,  —  3,  —  2,  —  1,  0,  1,  2,  3,  4,  5,  ?/  equals 
respectively  0,  ±3,  ±4,  ±4.5,  ±4.9,  ±5,  ±4.1),  ±4.5,  ±4,  ±3,  0. 

Locating  the  points  (—5,  0),  (—4, 
±3),  (  —  4,  —3),  etc.,  and  joining,  we 
obtain  the  graph  (a  circle)  ABC  of  the 
equation  x^  +  y'^  =  25. 

Locating  two  points  of  equation  ( 2) , 
e.g.  (—2,  0)  and  (0,  3),  and  joining  by 
a  straight  line,  we  obtain  DE,  the  graph 
ofSx-2y  =  -6. 

Since  the  two  graphs  meet  in  two 
points  P  and  Q,  there  are  two  pairs  of 
roots,  \vhich  we  find  by  measurement, 
x=1i,y  =  4^,0Tx=-4,y=-S. 

Ex.  2.    Solve  graphically  the  following  system : 

r       xy  =  12,  (1) 

U-?/=   2.  (2) 

12 
From  (1)  y  =  —  -     Hence  by  substituting  for  x  the  values  —  12,  —  11 


-^'^2^ 

A    ^"t^^^ 

t     y   SB 

7     i       t 

t    ^        4 

X'               7      0                    X 

-5-4  -3  y2    -1            12     3     4  5' 

tv    4        7 

v^     3        / 

"57^    'l         7 

^<i*  ^^c 

ir'Y' 

to  +  12,  we  obtain  the  following  points  :   (■ 


(-10,  -H),  (-9,  -10,  (- 
-2§),  (-4,  -3),  (-3,  -  4), 
12),  (2,  6),  etc.,  to  (12,  1). 


H). 


(-2, 


(-7, 
6),   (■ 


12,-1),  (-11,  -1,V), 
-If),  (-6,  -2),  (-6, 
■  1,  -12),    (0,  ±0)),  (1, 


GRAPHIC  REPRESENTATIONS 


28T 


Y  '      C                                         -1— 

i:            ■     tr 

z 

^z 

i-                     z 

V            -      ^ 

A                   Z        71 

V           j^          4- 

t      '  ^ 

^      D      ^ 

SP/                    TT 

2^              D 

V'                                                  r^             K        n T-W 

X                                  0       7 

^    __       _    __    _    _-_,^_    _         ^ 

5-                      ^X        ^ 

nN     ^ 

P  K           77 

'^^            4i_r-+-          -B'^ 

-^     \               ^            -52 

^       V              ^s,   T  ^ 

7                                                                ^^3^ 

7^              '-            ~^           7    '^^s   D- 

X  z                            y       si 

A  ^            c^          Ty;^ 

~vL             -t   v^      ^ 

ih                  ^    -Yf     -2':5  7 

Locating  these  points  and  joining  them  produces  the  graph  of  (1), 
which  consists  of  two  separate  branches,  CD  and  EB\ 

Locating  two  points 
of  equation  (2)  and 
joining  by  a  straight 
line,  we  have  the  graph 
AB  of  the  equation  (2). 

The  coordinates  of 
the  two  points  of  inter- 
section P  and  P'  are 
the  required  roots.  By 
actual  measurement  we 
find  x  =  4.5+,  ?/  =  2.5+, 
or  ic  =  —  2.5,  y  =  —  4.6. 

To  obtain  a  greater 
degree  of  accuracy,  tlie 
portion  of  the  diagram 
near  P  is  represented 
on  a  larger  scale  in  the 
small  diagram.  Since 
the  small  part  of  CD 
which  is  represented  is 

almost  a  straight  line,  it  is  sufficient  to  locate  2  or  3  points  of  this  line. 
By  actual  measurement  we  find  : 

x=      4.606,  y=      2.606. 

Evidently  the  second  pair  is 

x=  -  2.606,  y=  -  4.606. 

By  increasing  the  scale  further  and  further,  any  degree  of  accuracy 
may  be  obtained. 

EXERCISE  116 
Solve  graphically  the  following  simultaneous  equations : 

2a;-32/  =  6.  '     1     x  +  by=   6. 

^      r3:^  +  42/  =  10,  .    ^      rdx  +  ^y  =  l, 

14  a;  +    2/=   9-  *     \5x—    y  —  1, 

^      {2x-^y==      7, 

l3a?  +  22/=-8. 


4.5 


288 


ELEMENTARY  ALGEBRA 


6.    Show  graphically  that  the  following  system  cannot  have 
finite  roots ; 


(2x-y  =  2, 


x  —  y  =  4:. 

7.  What  are  the  relative  positions  of  the  graphs  of  two  linear 
inconsistent  equations  ? 

8.  Show  graphically  that  the  following  system  is  satisfied 
by  an  infinite  number  of  roots : 

(3x-2y  =  3, 

l6ic  —  42/  =  6. 
Solve  graphically : 

^       x^  +  /  =  16,  ^^     f     ^y  =  12, 


X  +y  =   2.  (x-y=   1. 

2'-y  +  l  =  0, 


x-\-y  +  l  =  0.  [x-  +  y^  =  25. 


X  =2y. 


Table  of  Squares,  Cubes,  and  Square  Roots 


X 

a;2 

0.3 

v'x 

X 

a;2 

tcs 

>/x 

1 

1 

1 

1,000 

21 

4  41 

9  261 

4,583 

2 

4 

8 

1,414 

22 

484 

10  648 

4,690 

3 

9 

27 

1,732 

23 

5  29 

12  1()7 

4,71^ 

4 

l(i 

64 

2,000 

24 

5  76 

13  824 

4,89i) 

5 

25 

125 

2,236 

25 

6  25 

15  625 

5,000 

6 

36 

216 

2,449 

26 

6  76 

17  576 

5,099 

7 

49 

343 

2,646 

27 

7  29 

19  683 

5,196 

8 

(54 

512 

2,828 

28 

7  84 

21  952 

5,292 

9 

81 

729 

3,000 

29 

8  41 

24  389 

5,385 

10 

100 

1000 

3,162 

30 

9  00 

27O00 

5,477 

1  1 

121 

1331 

3,317 

31 

9  61 

29  791 

5,568 

12 

144 

1728 

3,461 

32 

10  24 

32  7(^8 

5,657 

13 

169 

2  197 

3,()0f) 

33 

10  89 

35  937 

5,745 

14 

196 

2  744 

3,742 

34 

11  5() 

39  304 

5,831  . 

15 

2  25 

3  375 

3,873 

35 

12  25 

42  875 

5,916 

16 

2  56 

4  096 

4,000 

36 

12<)6 

46  ().-() 

6,000 

17 

2  89 

4  913 

4,123 

37 

13  69 

50  653 

6,083 

18 

3  24 

•   5  832 

4,243 

38 

14  44 

54  872 

6,164 

19 

3  61 

6  859 

4,359 

39 

15  21 

59  319 

6,245 

20 

4  00 

8  000 

4,472 

40 

16  00 

64  000 

6,325 

CHAPTER  XVIII 

QUADRATIC   EQUATIONS  INVOLVING  ONE  UNKNOWN 
QUANTITY 

315.  A  quadratic  equation,  or  equation  of  the  second  degree, 
is  an  integral  rational  equation  that  contains  the  square  of  the 
unknown  number,  but  no  higher  power;  e.g.  a^— 4a;  =  7, 
62/^  =  17,  aa^  +  6x-  +  c  =  0. 

316.  A  complete,  or  affected,  quadratic  equation  is  one  which 
contains  both  the  square  and  the  first  power  of  the  unknown 
quantity. 

317.  A  pure,  or  incomplete,  quadratic  equation  contains  only 
the  square  of  the  unknown  quantity. 

aa;2  4-  6a;  4-  c  =  0  is  a  complete  quadratic  equation. 
aa;2  =  ?7i  is  a  pure  quadratic  equation. 


PURE   QUADRATIC   EQUATIONS 

318.    A  pure  quadratic  is  solved  by  reducing  it  to  the  form 
a^  =  a,  and  extracting  the  square  root  of  both  members. 

Ex.  1.    Solve  13  a;2  - 19  =  7  ar  +  5. 

Transposing,  etc.,  6  x^  =  24. 

Dividing,  x^  =  A. 

Extracting  the  square  root  of  each  member, 
a;  =  +  2  or  x  =  -2. 
This  answer  is  frequently  written  a;  =  ±  2. 

Check.  13(± 2)-^- 19  =  3.3;  7(i2)2  +  6  =  33. 

u  289 


290  ELEMENTARY  ALGEBRA 

Ex.2.   Solve    <L:z^  =  ^±^. 
a-\-x     x  —  4,  a 

Clearing  of  fractions,  ax  —  ofi  —  4l  a^ -\- 4t  ax  =  ax  +  A  a^  +  x'^ -\- ^  ax. 
Transposing  and  combining,  _  2  a;2  =  8  a^. 

Dividing  by  -  2,  a;2  _  _  4  ^^2, 


Extracting  the  square  root,  ic  =  ±  V—  4  a^^ 

or  x=±2a^'-\. 

EXERCISE   117 
Solve  the  following  equations  : 

1.  17  0^2-7  =  418. 

2.  13a?2-19  =  7a^  +  5. 


9.    -^^— +  — ^^  =  2|. 
x-{-l      x  —  1 

ihx    810  1^-  ::r^  +  r-r^  =  ^i- 


2        3a; 
aa;2-6  =  c.  *"•    7a;-l~5a;-3 


11. 


a?  — 3     a?  +  3 

4a?  +  5       x-\-2 


5.    (a;  +  i)(^-i)  =  TV  12.    V23a;2-70a;+81  =  5a^-7 


6.  a;2  +  7  =  4.  13.    3— V7ir^— 24.i;+45=4ic. 

7.  4a;(a;-.l)=-4(a;-2).        14.  ^       ^Va;-4 

Va;4-4  ^ 

aHi2^-^__^4_  3 

•    a;-2      a;  +  2      x^-4.  15.    VV5  a;^  +  9  - 19  =  2. 


16.  2  +  Vl7-2a;=V2a;  +  17. 

17.  {a  +  x){h  —  x)-\-{a  —  x){h  +  x)=0, 

18.  (a  +  5rc)2  +  (aa;-?;)2  =  2(aV-&2). 

a  +  6a;     c  +  dx 

20.  If  a^  _|_  52  _  ^2^  f^^^  ^  1^  terms  of  h  and  c. 

21.  If  d  =  ^f,  solve  for  ^. 

22.  If  2  a^  +  2  52  ==  4  ^2  ^  ^2^  gQi^e  f^j,  r^^ 


QUADRATIC  EQUATIONS  291 

23.  Solve  the  equation  of  the  preceding  example  for  c. 

24.  If  a^  =  6^  +  c-  —  2  cp,  solve  for  b. 

25.  If  S  =  xr^,  solve  for,-.  ^g     jf  ij  =  ^',  golve  for  d 

26.  If  /S  =  4  irir,  solve  for  ?\  <^" 

27.  If  Z:^  =  «2.^^  solve  for^.       29.    If  ^  =  —,  solve  for  u 

EXERCISE   118 

1.    Find  a  positive  number  which  is  equal  to  its  reciprocal. 
'2.    The  ratio  of  two  numbers  is  5:4,  and  their  product  is 
980.     Find  the  numbers. 

3.  Three  numbers  are  to  each  other  as  2  :  3  :  4,  and  the  sum 
of  their  squares  is  261.     Find  the  numbers. 

4.  Two  numbers  are  as  5:4,  and   the  difference  of  their 
squares  is  36.     Find  the  numbers. 

5.  The  sides  of  two  square  fields  are  as  7 :  24,  and  they  con- 
tain together  10,000  square  yards.     Find  the  side  of  each  field. 


319.    A  right  triangle  is  a  triangle,  one  of 
whose   angles   is   a   right    one.      The    side    ^ 
opposite  the  right  angle  is  called  the  hypote- 
nuse (c  in  the  diagram).     If  the  hypotenuse       '  &" 
contains  c  units  of  length,  and  the  two  other  sides  respectively 
a  and  b  units,  then             ^.2  _  ^2  _|_  ^2 

Since  such  a  triangle  may  be  considered  one  half  of  a  rec- 
tangle, its  area  contains  ^  square  units. 


6.  The  hypotenuse  of  a  right  triangle  is  15  inches,  and  the 
two  other  sides  are  as  3  :  4.     Find  the  sides. 

7.  The  hypotenuse  of  a  right  triangle  is  to  one  side  as  41 :  9, 
and  the  third  side  is  40  centimeters.  Find  the  unknown  sides 
and  the  area. 


292  ELEMENTABY  ALGEBRA 

8.  The  hypotenuse  of  a  right  triangle  is  2,  and  the  other 
Wo  sides  are-  equal.     Find  these  sides. 

9.  The  area  of  a  rectangle  is  1260  square  feet,  and  the  two 
sides  are  as  5 :  7,     What  is  the  length  of  the  sides  ? 

iO.  The  area  of  a  right  triangle  is  147  square  feet,  and  the 
two  smaller  sides  are  as  2  :  3.     Find  the  sides. 

11.  The  area  of  a  circle  exceeds  the  area  of  another  circle 
by  616  square  inches,  and  their  radii  are  as  25  :  24.  If  the 
value  of  IT  is  assumed  equal  to  ^i^-,  find  the  radii  of  the  circles. 
(The  area  ^S"  of  a  circle  whose  radius  is  R  is  determined  by  the 
formula  S  =  ttE^) 

12.  Two  circles  together  contain  44,506  square  incheSj  and 
their  radii  are  as  15  :  8.     Find  the  radii. 

COMPLETE  QUADRATIC  EQUATIONS 

320.  Method  of  completing  the  square.  The  following  ex- 
ample illustrates  the  method  of  solving  a  complete  quadratic 
equation  by  completing  the  square. 

Solvea;2-7«  +  10=0. 
Transposing,  x-  —  7  x=  —  10. 

The  left  member  can  be  made  a  complete  square  by  adding 
another  term.  To  find  this  term,  let  us  compare  ix^  —  7  x  with 
the  perfect  square  x^  —  2  mx-{-  w?.  Evidently  7  takes  the  place 
of  2  m,  or  m  =  J.  Hence  to  make  cc^—  7i»  a  complete  square 
we  have  to  add  (|)^  which  corresponds  with  m^. 

Adding  (J)^  to  each  member, 

or  (^-|y  =  J. 

Extracting  square  roots,  a;  —  J  =  ±  f . 
Hence  a;  =  J  ±  |. 

Therefore  cc  =  5  or  oj  =  2. 

Check.  52-7.5+10  =  0,   22-7-2  +  10  =  0. 


QUADRATIC  EQUATIONS  293 

Ex.1.    Solve9x2-2  =  15a;-f 4. 

Transposing,  9  a;^  _  15  ^  =  6. 

Dividing  by  9,  aj'^  —  |  aj  =  f . 

Completing  the  square  (i.e.  adding  (|)2  to  each  member), 

Simplifying,  (x  -  f )2  =  |f. 

Extracting  square  roots,       x  —  f  =  ±  |. 
Transposing,  x  =  |  ±  |. 

Therefore  x  =  2,  or  —  ^. 

321.   Hence  to  solve  a  complete  quadratic : 

Reduce  the  equation  to  the  form  x^  -\-px  =  q.  Complete  the 
square  by  adding  the  square  of  one  half  the  coefficient  of  x.  Ex- 
tract the  square  root  and  solve  the  equation  of  the  first  degree  thus 
formed. 

Ex.2.   Solve^^  +  ?^^i=2. 
a  X 

Clearing  of  fractions,   x^  —  x -{- 2  a^  +  a  =  2  ax. 

Transposing,  x^  —  x  —  2  ax  =  —  2  a^  —  a. 

Uniting,  x'^  -  x(l  +  2  a)  =- 2  a^  -  a. 

Completing  the  square, 

Simplifying,               (._Li^«y  =  L:^. 
Extracting  square  root,  x —^ — -  =  ±  -Vl  —ia^. 

Transposing,  ^  _  1  +  2  a  _^  1  ^^  _  ^  ^g^ 

2  2t 


Therefore  ^^1  +  2a  ±  Vl -4a^ 

% 


294  ELEMENTARY  ALGEBRA 

Ex.  3.    Solve  a'x" -a^ -^2  a'x  + a^  =  0. 

Transposing,  a'^x'^  —  x'^  +  2  q?x  =  —  a\ 

Uniting,  x'^{a^  -\)  ■\-'la^x=-  a\ 

Dividing,  a;2  ^  -l^x  =  -  --^. 

Completing  the  square, 

Simplifying,  /'a;  +  —^—\  ^  = ^^ 

Extracting  the  square  roots, 


a2  _  1         ^2  _  1 

Transposing,  »  =  -  -^  ±  — ^. 

Therefore 

a;  =  -«i±«  = ^orx  =  -^'  +  ^  = ^. 

a? -I         a-\  a^-l  a  +  \ 

EXERCISE  119 

1.  a.-2  4-8a.-  =  33.  13.  10  a^  =  27  -  39  a;. 

2.  a^-6a;=7.  14.  28  x^  -  65  =  17  a;. 

3.  a.'2-6aj-16  =  0.  15.  27  a;^  _  30  a;  =  77. 

4.  a^  +  12a;  =  64.  le.  35 a^ - 65  =  66  a;. 

5.  a;2  4.ii,=  i2a;.  ^^^  8  a;^  _^  4I  ^  _l_  35  ^  0. 

6.  ar^-21  =  4a;.  ^g.  12  a;2  +  77  aj  +  30  =  0. 

7.  a;2-llx  =  60.  •       19^  a;(a;  +  5)-84  =  0. 

8.  a^-8a;  +  15  =  0.  g^.  3  a^- 3  a.-330  =  0. 

9.  a^  +  15  =  12a..  ^^  2.^-5.-3  =  0. 

10.  a^  +  17.  =  -70.  ^^     6.^  +  1  =  5.. 

11.  12a^  +  aj  =  35.  4„  ,  7 

2"?      7.4-1=—— I—. 

12.  15a^  =  7a?H-88.  ^        16. 


QUADRATIC  EQUATIONS  295 

24.  9^  =  x  +  i.  ^g    3x  +  _2I-  =  9. 

g  5a?  — 1 

25.  x  +  -  =  5. 

30.    ll.--i?-  =  16. 
18      .  3a;  +  l 


26.    a; =  3. 

X 


31.    3a;2_f a;+i-  =  0. 


27.    3a;-  — =  8.  32.    (3  a;  -  5)  (7  a;  -  8)  =  140. 


a; 


28.    5a;-— ?^  =  18. 


33.    (2a;-15)(3a;+8)=-154. 
2a;  +  3      ~"  34.    (a;  +  3)^  +  (x 4- 5)^  =  514. 

35.  (2a;  +  l)2+(3a;  +  l)'=(2a;  +  3)2. 

36.  S(x'-r)-24:  =  4.(x  +  5)(x-3). 

37.    -^—  =  ^^.  46.    _^  +  -A_  =  3. 

2(a;-3)      a;-l  a;-4      a;-3 

38      '^a;-5  ^5a;-3  ^^     _9 1_=^5 

10a;-3      6a;  +  l*  *   x-2     x-7 

39.  ^Ilg=    ^-^    .  48.    -^ 5-  =  9, 

a;  +  2      2a;  +  10  a;-2     x-6 

40.  ^^:il+^^=:l=?+a;-l.   49.    ^  +  ^  =  23. 

9  5         a;  a;  — 4     a;— 2        * 

50.  2^  +  3_^+2^3 
a;-3       a;-2 

51.  ^^:z^_I^±2     8  =  0 
=  3.  2a;-3      5a;-4^ 

52.  x^-\-2ax  =  Sa\ 

53.  x^-\-llm^  =  12mx. 

54.  3a;2_2yi2^53yyj^ 

55.  2x^-Bax  =  3a\ 
56.    a;2^(a  +  5)a.==2a2-5a6  +  262. 


41. 

a;24-22(a;  +  5)  =  0. 

42. 

^ay^-x  =  ^. 

43. 

2x(2x-5)          2 
2a;-l         2a;-l 

44. 

^-'  =  x, 

x-^l 

45. 

x-1       1 

a;  — 13      a; . 

296  ELEMENTARY  ALGEBRA 

57.  x^  +  ab  =  ax  4- bx.  ■  1.1 

60.    a  +  x=-  +  -' 
a      X 

58.  a;  +  -  =  a4--.  a-\-x     b-\-x     5 

X  a  61.    — ^ — ■-] —  =  ^. 

b-\-x     a-\-x     2 

59.  a^-{a  —  xy=(a  —  2x)K       62.    aa^  +  6a;  +  c  =  0. 


322.   Solution  by  formula.     Every  quadratic  equation  can  be 
reduced  to  the  general  form, 

aa^ -\- bx -\- c  =  0. 

Solving  this  equation  by  the  method  of  the  preceding  article, 
we  obtain  

„_-b±-Vb^-4:aG 
X  =5 — • 

2a 

The  roots  of  any  quadratic  equation  may  be  obtained  by  sub- 
stituting the  values  of  a,  b,  and  c,  in  the  general  answer. 

Ex.1.    Solve6a;2  =  26a;-5. 

Transposing,  5  ic^  _  26  ic  +  5  =  0. 
Hence  a  =  5,  6  =  —  26,  c  =  5. 

Therefore  x  =  +  26  ^  V(- 26)-^  -  4^5^ 

10 

10  5 

Ex.  2.    ^olvQ  px^  —  p^x —■  X  =  —  p. 

Reducing  to  general  form,   px^  —  (p^  -\- 1)  x  -\-  p  =  0. 

Hence  a  =p,  6  =-  (i?^  +  1),  c  =p. 

Therefore  x  =  p'  +  l  ±Ap' +  ^)'zd^ 

2p 

^p^  +  l±(pljzJ)  =  ^or  1. 
2p  ^        i) 


QUADRATIC  EQUATIONS  297 

EXERCISE  120 
Solve  by  the  above  formula : 
1.   3x'2-10x  +  3  =  0.  14.   12a^  +  aJ  =  6. 

3.  a52-5a;  +  3  =  0.  16.  5  x  -  se"  =^ -- 50. 

4.  2a'2-7a;  +  3  =  0.  17.  3ar^-5a;  =  2. 

5.  3ar^-2  =  5.T.  18.  9  a;^  ^  j^j  ^^  3;Iq^ 

6.  15x2-53a;=42.  19.  a;2_^2a;  =  40. 

7.  8aj2-17ir=115.  20.  x2_|.a;^_j. 

8.  9x2-32a;-16  =  0.  21.  4a;2- 12  gx  +  9  5^=  0. 

9.  7a;2_-39^83,^  22.  6  ^^-49  mx  +  8  w2  =  0. 

10.  6a;2  =  37aj-6.  23.  6x^  +  5nx  =  6n\ 

11.  6a;2  =  a;  +  12.  24.  4  a^^  -  6  c?a;  =  4  d^. 

12.  20  =  a^  — a?.  25.  ar^-i-2aa;=6. 

13.  104-16a;  =  -6a^.  26.  ar^-2  aa?  + 6  =  0. 

27.  abx^-(a'  +  b')x  +  ab  =  0. 

28.  a;2  — 4ax  +  4a--62=:0. 

29.  x'-2ax  =  ^b'--a\ 

30.  a2a^-&V-4a6a;=:a2-6^ 

31.  mV  —  m  (a  —  6)  a;  —  a&  =  0. 

32.  (n  —J))  x^-{-  (p  —  m) x-{-(m  —  n)  =  0. 

Find  the  roots  of  the  following  equations  to  two  decimal 
places : 

33.  a^  +  a;-l  =  0.  .37.  6.^2-5  3;  + 1  =  0. 

34.  2a^  +  2a?-l  =  0.  ^^'  -2 ar^ - .5 a;  =  .3. 

35.  .r^-4a:  +  2  =  0.  39.  ^  +  -A_=9. 

36.  :r-3a;  +  l=0.  40.  9  ar' =  a;  +  f . 


298  ELEMENTARY  ALGEBRA 

323.  Solution  by  factoring.  Let  it  be  required  to  solve  the 
^1"^'*°"=  6x^  +  5  =  26  a,; 

or,  transposing  all  terms  to  one  member, 

Resolving  into  factors, 

{5x-l)(x-5)  =  0. 

Now,  if  either  of  the  factors  5x  —  l,  a;  —  5  is  zero,  the  product 

is  zero.     Therefore  the  equation  will  be  satisfied  if  x  has  such 

a  value  that  either  k        i      a  /^\ 

5  £c  —  1  =  0,         -  (1) 

or  ic-5  =  0.  (2) 

Solving  (1)  and  (2)  we  obtain  the  roots 
x  =  ^  or  x  =  5. 

324.  Evidently  this  method  can  be  applied  to  equations  of 
any  degree,  if  one  member  of  the  equation  is  zero  and  the  other 
member  can  be  factored. 

Ex.1.     Solve  a^  =  I^^±l^. 

2 

Clearing  of  fractions,  2x^  =  1  x^  +  15x. 

Transposing,  2  x^  -  7  x^  —  15  x  =  0. 

Factoring,  x  (2  x  +  3)  (x  -  5)  =  0. 

Therefore  x  =  0,  2x  +  3  =  0,  ora;-5  =  0. 

Hence  the  equation  has  three  roots,  0,  —  f ,  and  5. 

Ex.  2.     Solve  a^-3a;2_4a;  +  12  =  0. 

Factoring,  a;2  (x  -  3)  -  4  (x  -  3)  =  0. 

(x2  -  4)  (X  -  3)  =  0. 
Or  (X  -  2)  (X  +  2)  (X  -  3)  =  0. 

Hence  the  roots  are  2,  —  2,  3. 


QUADRATIC  EQUATIONS  299 

Ex.  3.     Solve  ar'  +  2ar^-8x-|-5  =  0. 

By  the  factor  theorem  we  find  that  x  =  1  satisfies  the  equation,  and  is, 
therefore,  a  root  of  the  equation. 

Dividing  by  ic  —  1  we  obtain  the  other  factor,  and  have 

(a;-l)(x2  +  3x-5)  =  0. 

Hence  a;  -  1  =  0,  or  a;2  +  3  a;  _  5  =  q. 

Solving  the  second  equation  by  the  formula, 

2 
Ex.4.     Solvea^  +  a'  =  0. 

Factoring,  (x  +  a)  (a;^  —  ax -\-  a^)  =  0. 

Hence  a;  +  a  =  0,  or  aj^  —  aa;  -f-  a^  =  0. 

Therefore  a;  =  —  a,  or 


a  ±  av—  3 


2 

Ex.  5.     Form  an  equation  whose  roots  are  4  and  —  6. 
The  equation  is  evidently  (x  —  4)  (a;  —  (—  G)  )  =  0. 
Le.  (X  -  4)  (a;  +  6)  =  0. 

Or  x2  +  2  X  -  24  =  0. 

325.  If  both  members  of  an  equation  are  divided  by 
an  expression  involving  the  unknown  quantity,  the  resulting 
equation  contains  fewer  roots  than  the  original  one.  In 
order  to  obtain  all  roots  of  the  original  equation,  such  a  com- 
mon divisor  must  be  made  equal  to  zero,  and  the  equation 
thus  formed  be  solved.     E.g.  let  it  be  required  to  solve 

If  we  divide  both  members  by  ic  —  3,  we  obtain  a;  +  3  =  5 
or  x  =  2.  But  evidently  the  value  a;  =  3  obtained  from  the 
equation  x  —  3  —  0  is  also  a  root,  for  a;^  —  9  —  5  (ic  —  3)  =  0,  or 
(a;  -  3)  (a;  +  3  -  5)  =  0.     Therefore  a;  =  3  or  a;  =  2. 


300  ELEMENTARY  ALGEBRA 

EXERCISE  121 

Solve  by  factoring : 

1.  x~-6x  =  7  12.  x^=:5x^-\-4.x. 

2.  a;2  +  8a;==20.  13.  (a?- l)(a;--12a;  +  27)=0. 

3.  x^-16  =  6x.  14.  x{x-'6)  =  7  x-^2. 

4.  x(x  +  5)  =  14:.  15.  a^-{-x  =  5x(2x-4), 

5.  x''-7x  =  S.  16.  a;^-16  =  0. 

6.  x^-9x-{-20  =  0.  17.  aj3  +  i«2-4a;-4  =  0. 

7.  «2  +  29aj  =  210.  18.  5  aj«-2a^- 10  a;  +  4  =  0. 

8.  ic3-lla;2-f  18aJ  =  0.  19.  Sa^  +  lSx'^SOx. 

9.  4a^4-17aj2  +  4a;==0.  20.  3a^  +  7ic  =  6. 

10.  iB«-8  =  0.  21.    6a;2  +  7a;  =  3. 

11.  x2(10a;  +  l)=3a;.  22.    6  a;(a;  -  3)  =  5(a;  -  3). 

23.  (a  — aj)(a;— 6)  =  (a  — x)(c  — ic). 

24.  a^-x^=(a-x)(b-\-c-x). 

25.  a;2=(a+6)a;.  27.    x(x^ -l)^5(x~l). 

26.  2i»3  +  a;^-5a;  +  2  =  0.  28.    a.-^- 6a;2  + 11  aj- 6  =  0, 

29.  flr^-4a^-20a;  +  48  =  0. 

30.  Find  all  roots  of  the  equation  ic^  =  1. 

31.  Find  all  values  of  -S/8. 
(Hint.     Let  x  =  \^,  hence  x^  =  8,  etc.) 

Solve: 

32.  -L--- i-  =  ^.  34.    -^4--^  =  -^. 

x+5     x-6     s-1  x-2     x-3     7-x 

33.  -l_  +  _^  =  -6__.  35.        7       ■       9  4" 


x  —  4:     x  —  3     x  —  6  x  —  3      x  +  o      x-\~l 


QUADRATIC  EQUATIONS  301 


36.   3a;-8  =  4V4a;  +  l*  39.   6 H-5vl3 a;  +  9  =  8a;. 


37.   4a;-3V7a;-6  =  6.  40.    Vaj4-5+Va;-3  =  4. 


38.    5  +  3^3x-5=2x.  41.    VaJ-3  +  Va;-8  =  5. 

42.    VaJ  +  7— Vic  — 6  =  2. 


43.    Vaj  +  2  +  ViC  —  3  =  V3a;  +  4 


44,    Va;  +  7  + V2a;  +  7  =  V8a;  +  9. 


45.    V3  «  4-  4  —  V6  a?  — 11  =  Vic  —  3 

a;  +  <^     a;  — g  _  q  i  ^^       36   _  2a   __  2a  +  & 

'a;  —  a     a;  +  a  *a;  —  a     a?— 6       a  +  a 

48.    ^llM±l  =  ,_3. 
a^-6a;-f-9 

40    a?  — «,a;  +  &_      &       ,      ce—b 
*    a-6       2a   ~2a-a;     2a-26' 


60.    x  —  b-y/x=:a(a  —  b).  Va  +  ^4- Va— a;     a 

53.  —    ... .  ■■■    =c  — * 

--     aa;  +  6      mx  —  n  -Va-^-x  —  va  —  x     ^ 

51.     ; = • 

bx-\-a     nx  —  m 

52.  ex+-^  =  (a+by.       54.  JiEi+./*±i=c+i. 

a  +  6  >'&+a;      ^a— a?  c 

Form  the  equations  whose  roots  are : 

55.  2,1.  57.    -4,-5.  59.    1,2,3. 

56.  2,-3.*  58.    0,3.  60.    -1,2,0. 


PROBLEMS  INVOLVING  QUADRATICS 

326.  Problems  involving  quadratics  have  in  general  two 
answers,  but  frequently  the  conditions  of  the  problem  exclude 
negative  or  fractional  answers,  and  consequently  many  prob- 
lems of  this  type  have  only  one  solution. 

*  The  answers  of  radical  equations  must  be  substituted  in  the  original 
equation  in  order  to  determine  which  roots  are  true  roots.     (See  §  285.) 


302  ELEMENTARY  ALGEBRA 

EXERCISE  122 

1.  The  square  of  a  number  increased  by  14  times  the  num- 
ber equals  275.     Find  the  number. 

2.  What  number  increased  by  its  reciprocal  equals  ^  ? 

3.  The  difference  of  two  numbers  is  49,  and  their  reciprocal 
values  differ  by  |-.     Find  the  numbers. 

4.  What  number  exceeds  its  reciprocal  by  w  ?     Find  the 
number  if  (a)  w  =  |,  (6)  n  =  |. 

5.  Find  two  numbers  whose  difference  is  7,  and  whose 
product  is  30. 

6.  Find  two  factors  of  1728  whose  sum  is  96. 

7.  Find  two  consecutive  numbers,  the  sum  of  whose  squares 
equals  113. 

8.  A  man  sold  a  horse  for  $  96,  and  gained  as  many  per 
cent  as  the  horse  cost  dollars.     Find  the  cost  of  the  horse. 

9.  A  man  sold  a  watch  for  $  16,  and  lost  as  many  per  cent 
as  the  watch  cost  dollars.     Find  the  cost  of  the  watch. 

10.  A  needs  3  days  more  than  B  to  do  a  certain  piece  of 
work,  and,  working  together,  the  two  men  can  do  it  in  2  days. 
In  how  many  days  can  B  do  the  work  ? 

11.  A  cistern  is  filled  by  two  pipes  in  40  minutes,  and  the 
larger  pipe  can  fill  it  in  one  hour  less  time  than  the  smaller 
one.  In  how  many  minutes  can  the  smaller  pipe  fill  the 
cistern  ? 

12.  A  number  of  boys  bought  a  boat,  each  paying  as  many 
dollars  as  there  were  boys.  Had  there  been  8  boys  more,  each 
would  have  paid  $  4  less.  How  many  dollars  did  each  boy 
pay? 

13.  If  a  train  had  traveled  5  miles  an  hour  faster,  it  would 
have  needed  one  hour  less  to  travel  150  miles.  Find  the  rate 
of  the  train. 


QUADRATIC  EQUATIONS  303 

14.  A  wheelman,  traveling  a  distance  of  72  kilometers, 
would  arrive  30  minutes  earlier  if  he  traveled  2  kilometers 
per  hour  faster.     Find  his  rate  of  traveling. 

15.  To  pave  a  room  with  square  tiles  of  a  certain  size,  360 
tiles  are  needed.  If  each  tile  were  one  inch  longer  and  one 
inch  wider,  250  tiles  would  be  needed.  Find  the  dimensions 
of  the  tiles. 

16.  A  rectangular  field  has  an  area  of  180,000  square  feet, 
and  a  perimeter  of  1720  feet.  Find  the  dimensions  of  the 
field. 

17.  Find  the  price  of  an  egg,  if  1  more  for  6^  would  dimin- 
ish the  price  of  100  eggs  by  $  1. 

18.  One  side  of  a  rectangular  field  exceeds  the  other  side  by 
70  feet,  and  the  area  equals  49,400  square  feet.  Find  the 
dimensions. 

19.  On  the  prolongation  of  a  line  AC,  10  inches  long,  a 
point  B  is  taken,  so  that  the  rectangle,  con- 
structed with  AB  and  CB  as  sides,  contains  A  C  B 
39  square  inches.     Find  AB  and  CB. 

20.  A  line  AB,  20  inches  long,  is  divided  into  two  parts,  AC 
and  CB,  so  that  AC  is  the  mean  proportional  between  CB  and 
AB.  Find  the  length  of  AC.  (AB  is  said  to  be  divided  in 
"  extreme  and  mean  ratio.") 

21.  Solve  the  preceding  problem  if  AB  equals  a  inches. 

22.  A  line  AB  is  divided  into  two  parts,  AC  and  CB,  so  that 
^C  is  the  mean  proportional  between  CB  and  AB.  If  AC 
equals  1  inch,  find  AB. 

23.  The  length  AB  of  a  rectangle,  ABCD,  exceeds  its  width 
AD  by  119  feet,  and  the  line  BD  joining   a 
two  opposite  vertices  (called  "  diagonal ") 
equals  221  feet.     Find  AB  and  AB.  q 


304  ELEMENTARY  ALGEBRA 

24.  A  rectangular  grass  plot,  40  feet  long  and  30  feet  wide, 
is  surrounded  by  a  walk  of  uniform  width.  If  the  area  of  the 
walk  is  I  of  the  area  of  the  plot,  how  wide  is  the  walk  ? 

25.  A  circular  basin  is  surrounded  by  a  path  6  feet  wide,  and 
the  area  of  the  path  is  J  of  the  area  of  the  basin.  Find  the 
radius  of  the  basin. 

26.  A  number  is  formed  by  two  digits,  and  the  tens  digit 
exceeds  the  units  digit  by  2.  If  the  number  is  divided  by  the 
product  of  its  digits,  the  quotient  is  2|.     Find  the  number. 

27.  A  man  bought  a  certain  number  of  pounds  of  tea  and  10 
pounds  more  of  coffee,  paying  $  14.00  for  the  tea  and  $  9.00  for 
the  coffee.  If  a  pound  of  tea  cost  40  ^  more  than  a  pound  of 
coffee,  what  was  the  price  of  the  coffee  per  pound  ? 

28.  A  stone  is  dropped  into  a  well,  and  the  sound  of  its 
impact  upon  the  water  is  heard  at  the  top  of  the  well  4|  seconds 
later.  If  the  velocity  of  sound  is  assumed  as  360  meters  per 
second,  and  g=^10  meters,  how  deep  is  the  well  ?     (A  body  falls 

in  t  seconds  ^t^  meters.) 

EQUATIONS  IN  THE  QUADRATIC  FORM 

327.  An  equation  is  said  to  be  in  the  quadratic  form  if  it 

contains  only  two  unknown  terms,  and  the  unknown  factor. of 
one  of  these  terms  is  the  square  of  the  unknown  factor  of  the 
other,  as 

328.  Equations  in  the  quadratic  form  can  be  solved  by  the 
methods  used  for  quadratics. 

Ex.1.    Solve  x^-9a^-\-S  =  0. 


By  formula,  ^^1±VEZ^ 

2 

=  9±I  =  8,orl. 


QUADRATIC  EQUATIONS  305 

Therefore  x  =  v^8  =  2,  or  a;  =  v/T  =  1. 

If  the  factoring  method  is  used,  we  have 

(a;3-8)  (x^-1)  =  0. 
Writing  each  factor  equal  to  zero,  and  solving, 

x  =  2,  x=-l± V^^,  x  =  l,  x=~'^^-^~'^ 


2 

Note.     The  solution   by  means  of   factoring  frequently  produces  all 
roots,  while  other  methods  produce  only  some  roots  of  the  equatiou. 

-    Ex.2.    Solve  x~^  -  33  a;"5  +  32  =  0. 

Factoring,  (x~^  -  32)  (x"^  -  1 )  =  0. 

Therefore  x~^  =  82,  or  1. 

Raising  both  members  to  the  -f  power, 

jc  =  32~^  or  r^^jJj  orl. 

EXERCISE  123 
Solve  the  following  equations : 
1.    x^-ox'  =  36.  13.    a;_5V^+6  =  a 

— '  14.    x^  -\-6  x^  =  10. 
-^B.    var  — 2va;  =  24. 
-46.    3vV  +  2-C/a;-16  =  0. 


17.  ar''»-3x"  +  2  =  0. 

18.  a;^-2ax2  +  a2-62  =  o. 

19.  x^-{-6x~^  =  5. 


2n  2n 


6.  (a;2-10)(ar'-3)  =  78. 

.  8.  x-*  +  16  =  17x-^. 

9.  8V^  =  15a;-^-14.  20.  3  a;^  +  2  =  16  a;" 3. 

10.  3  ic^  +  3  a;"^  =  10.  21.  Va;-l  =  a;-l. 

11.  a;-2V^  — 3  =  0.  22.  x—(a-\-b)Vx=2 a{a—b). 

12.  a;  +  3V^  =  10.  23.  v/»  +  V^=20. 

24.    x-i-ab  =  (a-^b)Vx  +  2(a-by, 


f^^^r-^i^    ^/3, 


306 


ELEMENTARY  ALGEBRA 


329.   In  more  complex  examples  it  is  advantageous  to  sub 
stitute  a  letter  for  an  expression  involving  x. 


{    X    J    '^-        X 

Let 

X 

Then 

?/2+15=8y, 

or 

2/2-8^+15=0, 

or 

(2/-6)(2/-3)=0. 

Hence 

y  =  6,  or  y  =  3. 

Le. 

^  +  4  =  5, 

X 

or   ^  +  4^3. 

Solving, 

x-\-4  =  5x, 

x  +  4  =  Sx. 

4x  =  4, 

2x  =  4. 

x  =  l, 

x  =  2. 

Ex.2.     Solve   x^-Sx-2Vx^-Sx  +  A0  =  -5. 

r2 


Adding  40, 

Let 

Hence 


Therefore 


£c-  -  8  X  +  40  -  2  Vic2  -  8  X  +  40  =  35. 
Vx2  -  8  X  +  40  =  y,  then  x^  -  8  x  +  40  =  2/2. 
?/2-2?/  =  35. 
2,2  _  2  y  _  35  =  0. 
(y-7)(2/  +  5)=0. 


Vx2' 


8x  +  40  =  7, 


or  y  =  —  5. 


x2  -  8  X  +  40  =  49, 
x2  -  8  X  -  9  =  0, 
(a;_9)(x  +  l)=0, 
X  =  9,  or  —  1. 


or  Vx^  -  8  X  +  40  =  -  6. 
x2  -  8  X  +  40  =  25. 
x2-8x  +  15=0. 
(x  -  5)  (x  -  3)  =  0. 
X  =  5  or  3. 


Since  both  members  of  the  equation  were  squared,  some  of  the  roots 
may  be  extraneous.  Substituting,  it  will  be  found  that  9  and  —  1  satisfy 
the  equation,  while  5  and  3  are  extraneous  roots. 

This  can  be  seen  without  substituting,  for  5  and  3  are  the  roots  of  the 
equation  Vx^  —  8  x  +  40  =  —  5.  But  as  the  square  root  is  restricted  to  its 
positive  values,  it  cannot  be  equal  to  a  negative  quantity. 


QUADRATIC  EQUATIONS  307 


EXERCISE   124 


1.   3(ar^-2)2  +  6(a;2-2)  =  2.4.   ^     a^  +  1     x'-l^U^ 
f       -j\2        /        -\\  ar  —  l      ar  +  l      15 

3.   .^  +  5.-5  =  ^^.  ,.   ^  +  2.-5=      24 


a^  +  2x 


(-1 


;Y  +  3aj  +  ^-  =  4.  8.   a^-6a;  +  12=       ~  ^^ 


xj                X  a^  —  6a;  +  l 

9.    a^_4a;-3  =  --— i? 

10.  («2_5x  +  2)2  +  6(a;2_5a;  +  2)  +  8  =  0. 

11.  (a-2  +  3a;-2)2-4(a^  +  3aj-2)  =  32. 

12.  2(x'  +  2x  +  3y-(x'  +  2x-\-4:y  =  9S. 

13.  (a^  +  2a;-l)(a;2  +  2a;-2)  =  2. 

14.  (or' -3)2 +  4(0^ -3)  =  5. 

15     a^  +  m^     a^-m2^41  a^  +  6       35a;  ^^p 

a^_m2"^a;2_^^^2     20'  *        X         a^  +  e 

17.    (3a;-2-2)2_5(3a;-2_2)  +  4  =  0. 

18.     v^   ^21 -y^^^ 

21 -V^  Va; 

19.    (a;  +  V^y-(a;4-V^)2  =  26,592. 


23.    a:2_6^2V^+9. 
+  ^        "         24.    «2  =  V^^^  +  13. 


25.  a;2_7_5Var^_7  =  24. 

26.  (a;  +  3)  +  (a;  +  3)^  =  2. 

^ 4  27.  a;-3-(a;-3)^  =  6. 

22.    V^M^  +  ^r.      ~  =  4.  ,_ 

Vic'  +  S  28.  4a;-7-2V4a;-7  =  15 


308  ELEMENTARY  ALGEBHA 


29.    2x^  +  ^x-ll^^■^/2^  +  6x-ll  =  2^. 


30.    4^x^  +  1  x  +  b  —  3  V4  a;2  4-  7  a;  _f-  5  =  4. 


31.   3a;2-2  =  10  +  3V3aj2_2. 


32.    2aj2_4a;-f 9=3  +  2v2i»^-4a;  +  9. 


33.   a^  +  i»  =  4Vi»2+a;  +  3  — 6. 


34.    x'^-\-x  —  %  =  Wx'^  +  x  +  ^. 


35.   3a^-4ir  +  2V3x2-4aj-6  =  21. 


36.    16aj-3V3a;--16a?  +  21  =  3aj2_7. 

^^-    \3a.-7^\2a.-5""^- 
a;^  +  a;  —  2     a^4-ic--4 

GRAPHIC   SOLUTION   OF   QUADRATIC   EQUATIONS* 

330.  Quadratic  equations  may  be  solved  graphically  by  the 
method  used  in  a  preceding  chapter.  (§  305.)  It  is,  however, 
simpler  to  solve  such  equations  by  the  method  employed  for 
the  solution  of  simultaneous  equations. 

Consider  the  equation        ax^ -\-hx-\-c  =  0.  (1) 

Let  y  =  ^.\  (2) 

Substituting  in  (1),  ay  -\-hx-\-  c  =  0.  \  (3) 

The  solution  of  the  system  (2),  (3)  for  x  produces  the  required 
root  of  (1). 

Tlie  advantage  of  this  method  lies  in  the  fact  that  the  graph 
of  the  linear  equation  (3)  is  easily  constructed,  while  the  graph 

*  This  section  may  be  omitted. 


QUADRATIC  EQUATIONS 


309 


of  (2)  is  identical  for  all  quadratic  equations.  Hence  the 
graph  of  the  equation  y  =  x^,  which  is  represented  in  the 
annexed  diagram,  may  be  used  for  the  solution  of  any  numeri- 
cal quadratic  equation,  provided  its  roots  lie  between  the  limits 
of  the  represented  abscissas  (—6  and  +6). 


Ex.  1.    Solve     11  a;2  +  30  cc  - 165  =  0. 

Let  y  =  x^. 

Then  11  y  +  30ic  -  165  =  0. 


(1) 

(2) 
(3) 


In  (3),  if  a;  =  0,  then  y  =  1B;  if  y  =  0,  then  x  =  5|.  The  straight  line 
joining  the  points  (0,  15)  and  (5^,  0)  is  the  graph  of  (3),  which  intersects 
the  graph  of  (2)  in  F  and  P'.     By  measuring  the  abscissas  of  F  and  P', 

we  have  x  =  2.7,  or  a;  ;=  -  5.5. 


Ex.2.    Solve     5a^-14a;-65  =  0. 


Let 
Then 


y  =  x'-*, 
5  V  -  14  x  -  G5  =  0. 


(1) 
(2) 
(3) 


310 


ELEMENTARY  ALGEBRA 


Locating  two  points  of  the  equation  (3),  e.g.  (0,  13)  and  (5,  27),  and 
joining  by  a  straight  line  produces  the  graph  of  (3),  which  intersects  the 
graph  of  (2)  in  Q  and  Q'.     Measuring  the  abscissas  of  Q  and  Q',  we 


obtain 


5.3,  or  ic  =  —  2.5. 


331.    In  the  equation  ay  -{-  bx  -\-  c  =^  0,  ii  x  =  0,  then  ?/  =  —  _, 

CL 

and  if  y  =  0,  then  x  =  —  -'     Hence,  by  taking  on  the  aj-axis 

c                      ^                                c  . 

the  point  —j-  and  on  the  ^/-axis  the  point ,  and  applying  a 

straight  edge,  the  roots  of  the  equation   ax-  +  bx-\-c  =  0  can 
frequently  be  determined  by  inspection.     In  some  cases,  how- 


J_L                          V,  ■  J 

L       __     _:       __ ;___^;^ _      _       __     __:       _: 

u                       _.           __        __     -35 

-V    :       ::             :     :      :  :    :±    :          :      :      ::     ::::     ^ 

A               -                                        x                                                    t. 

^   Vp^              -                                           -     ,.     -     ±                                 it                             -             - 

:».^___        on                                                                           J. 

^^■^^                                                                                                                                                         Je^' 

^^                                                                                                                                                  -'T 

K    rril  7                                                                        J-ri/^ 

--    --              "^L/iT                                        0^              ^^                              ^^      L 

:        r           <^n  "     "      ■  "~  -^---          —    -     ~~z^        ~~  " 

^-f    -      ^x0:^-           ~                                  xpv         t 

"                 \                                          ^S^   7                                                                                          .        -                     p^^j^''                           ]    " 

:           ^     "         ^   ^  ^6  r       ~                                 riE;S "               ^- 

\                       >^7^'         "^                 t'*--'^                 7  ' 

_               ^^              .           ^^^.^       -             ^    -         - 

::^^ ::::::::::::::: v;:::  +  ::::;^::::::  ::::::^|;::::::: 

:     :::::±^:  ::::::::::::::-2ii5::-f:::::::  i:::::^!::::::::: 

,1,         __      _     ''J^p-^      -4-         i 

:                   ^     "                 "        "^^^^                ±      z"        '        ' 

s,  :                 :     -,^-        "-^                     / 

^v-                    -   ^^   -              •-.                  / 

_        _        ^                               ^-^             Tn                 S                      ^ 

^_ ,             _       .1Q___^_^^                      _                 

-S              .i     -              -                                  ±     ^pl                      "I 

S>           ^^                                                                                                    ><s    ^ 

^V-^                                                                                                 <!^      ^..                                                   " 

"i?^               "                 R                          Z              i                  '     "      " 

;'Q   s "T --^t_  .__=.,_     __. 

-     ^'^-     -4-        ^^                                                     ^^                                  ^' 

-^         -                      ^^                     -                      ^^^                  .                         ^^. 

v-4-                  ^                                            ■"*                                  ^^                                                        "^^X 

X-                      ^''                                             -                  ""-"^fx^-"                                                              .              ^v    A. 

-v^           41'          -^           -7     '       -1        ""O      ^      ■     ■          ,            ■■■ 1'             \      'V> 

^^ 

V-'   i; 

IT  '^ 

ever,  it  is  more  advantageous  to  locate  one  or  both  points  out- 
side the  axes. 

Equations  of  the  third  and  higher  degree,  which  are  linear 
with  the  exception  of  one  term,  can  be  solved  by  the  same 
method. 

Note.  The  student  should  construct  the  graph  oi  y  —  x'^  on  a  large 
scale.  It  is  advisable  to  make  the  scale  of  the  ordinates  considerably 
smaller  than  the  scale  of  the  abscissas. 


QUADRATIC  EQUATIONS  311 

EXERCISE  125 
Solve  the  following  equations  by  the  graphical  method : 

1.  a^_a;-6  =  0.  14.    a.'2_2a:-9  =  0. 

2.  :x?  +  x-2  =  0.  15.    3aj2-f  7a;-42  =  0. 

3.  a^_3a;-18  =  0.  16..  2cc2  +  5aj- 20  =  0. 

4.  x2  +  3a;-10  =  0.  17.    if2  +  2x-10  =  0. 

5.  a;2_2^_8  =  0.  18.    5a;2-4a;- 5  =0. 

6.  aj2  +  2a;-4  =  0.  19.    5i«2  +  4a;- 25  =  0. 

7.  a:2_3^_-L2  =  0.  20.    ^x?  -  ^^x-^m  =  0. 

8.  a^  +  3a;-10  =  0.  21.    9a;2- 12.^•- 26  =  0. 

9.  ic2  +  a;-3  =  0.  22.    Occ^- 12a;- 100  =  0 

10.  x^-nx-W  =  Q.  23.  3a;2-2a;-65  =  0. 

11.  4a^-25a;  +  20  =  0.  24.  2a^  + cu-36  =  0. 

12.  3a;2  +  20a;  +  12  =  0.  25.  2^2  + 5a;- 12  =  0. 

13.  ^2_^a;-5  =  0.  26.  x'-Qx-\-^  =  0. 

27.    Has  x'^  -\-  x-\-\  =0  any  real  roots  ? 
28.    3a.-»-h5x-15  =  0.  29.    a.-^  -  5a;- 2  =  0. 


CHAPTER   XIX 
SIMULTANEOUS  QUADRATIC  EQUATIONS 

332.  The  degree  of  an  equation   involving   several  unknown 

quantities  is  equal  to  the  greatest  sum  of  the  exponents  of 
the  unknown  quantities  contained  in  any  term., 

ajy  +  2/  =  4  is  of  the  second  degree. 
x^y  +  5  aj2?/3  —  2/4  is  of  the  fifth  degree. 

333.  A  symmetrical  equation  is  one  wTiich  is  not  altered  by 
interchanging  the  unknown  quantities. 

x-\-y  =  xy^x^  -{■  x'^y^  +  ?/2  =  4  are  symmetrical  equations. 
x  —  y  =  2  and  x^  —  y^  =  1  are  not  symmetrical,  but  a  change  of  sign 
would  make  them  symmetrical. 

334.  A  homogeneous  equation  is  an  equation  all  of  whose 
terms  are  of  the  same  degree  with  respect  to  the  unknown 
quantities. 

4:x^  —  Sxh/  =  Sy^  and  x^  —  2xy  —  5y^  =  0  a^re  homogeneous  equations. 

335.  The  absolute  term  of  an  equation  is  the  term  which 
does  not  contain  any  unknown  quantity. 

Inx^  —  Axy  ■}■  2  =  0  the  absojute  term  is  2. 

336.  Simultaneous  quadratic  equations  involving  two  un- 
known quantities  lead,  in  general,  to  equations  of  the  fourth 
degree.  A  few  cases,  however,  can  be  solved  by  the  methods 
of  quadratics.* 

*  The  graphic  solution  of  simultaneous  quadratic  and  higher  equations 
has  been  treated  in  Chapter  XVII. 

312 


SIMULTANEOUS  QUADRATIC  EQUATIONS  313 


I.    EQUATIONS  SOLVED  BY  FINDING  x -^  y  AND  x-y 

337.  If  two  of  the  quantities  x-\-y,  x  —  y,  xy  are  given,  the 
tliird  one  can  be  found  by  means  of  the  relation  (x  -{-yy  —  4:xy  = 
{x-yf. 

(1) 

(2) 
Squaring    (1),  x"^ -\- 2  xy  -\- y"^  =  2b.  (3) 

(2)x4,  4xy  =  16.  (4) 

(3) -(4),        a:2-2xy  +  2/2  =  9. 
Hence  aj  —  y  =  ±  3.  -         (6) 

Combining  (5)  with  (1),  we  have 

or 
ic-?/  =  3.  x-y  =  -^. 


Ex.  1.     Solve 


\xy  =  4r. 


U=l. 


X  =  1, 

Hence  .|  '  or 

4. 


338.   In  many  cases  two  of  the  quantities  x-\-y,  x  —  y,  and 
xy  are  not  given,  but  can  be  found. 

j2x'-3xy  +  2f  =  8,  '  (1) 

^''■^-  1  0,-^=1.  (2) 

Square  (2),  x^  -  2xy  +  y^  =  1.  ,  (3) 

(3)  X  2,  2x2  -  4 xy  +  2 y2  =  2.  (4) 

(l)-(4),  xy  =  6. 

Hence  4xy  =  24.  (6) 

(3) +  (5),  x2_|.2xy  +  y2  =  25. 

or    X  +  ?/  =  —  5. 


Therefore  x  +  y  =  +5, 

But  X  —  y  =  1, 

Hence  x  =  3,  y  =  2, 

2  .  32  -  3  .  3  .  2  +  2  .  22  =  8, 

^""'*-  <  3-2=1. 


x-y  =  1. 

x  =  -2,  y  =  -3. 
r2.  22 -3. 2. 3  +  2.  32  =  8, 
I  -2  +  3  =  1. 


314 


ELEMENTARY  ALGEBRA 


339.    The  roots  of  simultaneous  quadratic  equations  must  be 
arranged  in  pairs,  e.g.  the  answers  of  the  last  example  are : 


x  =  S, 


or 


|2/  =  -3. 


EXERCISE  126 


Solve: 

1 


[xv  =  6. 


2. 


3. 


xy 

«^  -  2/  =  9, 
xy  =  36. 

x  +  y  =  2S, 


xy 


187. 


(x-y  =  5, 
\xu  =  176. 


5. 


6. 


7. 


xy 

X  -  2/  =  24, 
xy  =  4212. 

x  +  y  =  -6, 
xy  =  -  2592. 

x-y  =  Q>l, 
xy  =  876. 

50, 


i^  +  f-- 
[xy=7. 

(x'  +  f  =  S7, 
[x  +  y  =  7, 

^     (x'  +  f  =  lS, 
[x-\-y  =  5. 


11. 


x^  +  f  =  25, 
x-y=l. 


12 


13 


r  0^  +  2/'  =  100, 

Uv  =  48. 

■I 


14. 


15. 


16. 


xy 

x^-\-f  =  S2, 

x  +  y=^  —  S. 

^  +  xy  +  2/^  =  '^j 

x-y  =  l. 

x^-Zxy-\-f=-ll, 
a;2  +  2/'  =  34. 

x^y-\-xy  =  ll, 
x  +  y  =  Q,. 


17 


.    \x      y 


H- 


18. 


19. 


20. 


^  +  3a:?/  +  2/'  =  211, 

^+1  =  1 
13 

x-y  =  h, 

xy  =  a^-\-  ah. 

x-\-y  =  2a, 

.x'-\-y^  =  2a^  +  2h\ 


SIMULTANEOUS  QUADRATIC  EQUATIONS  315 


a.  ONE  EQUATION    LINEAR,   THE    OTHER    QUADRATIC 

340.  A  system  of  simultaneous  equations,  one  linear  and 
one  quadratic,  can  be  solved  by  eliminating  one  of  the  unknown 
quantities  by  means  of  substitution. 


7-3y_ 

2 


Ex.     Solve  2x-{-3y  =  7, 

From  (1)  we  have, 

Substituting  in  (2) ,        f'Lz3l\ ^^2y^-y  =  5. 

Simplifying,       49 -i2y -[-dy^ ■{- Sy^ -iy  =  20. 
Transposing,  etc. ,  ny^-46y  +  29=  0.. 

Factoring,  {y -l){17  y  -29)  =  0. 

Hence  y  =  1,  or  f  f . 

Substituting  in  (3),  x  =  2,  or  ^. 


(1) 
(2) 
(3) 


EXERCISE  127 


Solve ; 

I 


2. 


4. 


5. 


xy  —  5x  =  l, 
7  x  —  y  =  l. 

xy  =  6, 

2x  +  3y  =  lS. 

Qi^-\-4:xy  =  57, 
x  +  y  =  7. 

x^y  =  6, 
xy  =  16. 

2x-3y=l, 
5s^-7y^=13. 

5x^-}-y  =  3xy, 
2x-y  =  0, 


8. 


9. 


10. 


11, 


12. 


3x-y  =  0. 
x-3y-{-5=r0, 
y-xy  =  0. 
x^  +  y'  =  37, 
9x-^7y=61. 
x^-\-y^  =  7-{-xy, 
2x-3y  =  0. 

x-[-y  =  9. 
5x—y_       7 


4  4:X-{-3y 

3x-2  y  =  l. 


316 


ELEMENTARY  ALGEBRA 


13. 


14. 


3  +  4-^' 

3     4^3^ 
X     y     2 


4     5 


3, 


{x  +  yy  =  200-x. 


15. 


16. 


x:y  =  2'.S, 

'  y  +  8  ^  3^_+j/, 
a."  +  2      3x  —  y 

2x-y  =  2. 


III.   HOMOGENEOUS  EQUATIONS 

341.  If  one  equation  of  two  simultaneous  quadratics  is 
homogeneous,  the  example  can  always  be  reduced  to  an 
example  of  the  preceding  type,  for  one  unknown  quantity  can 
be  expressed  in  terms  of  the  other. 

Consider  the  homogeneous  equation, 

4.x'-llxy  +  6y^  =  0.  (1) 

Expressing  x  in  terms  of  y  by  means  of  the  formula  for 
quadratics, 

X 


lly±  V(ll  2/)' - 4  .  4  .  6 / 


8 

_lly±5y 
~        8 
=  2y,  or  I  y. 

In  most  cases  this  result  can  be  obtained  more  simply  by 
factoring,  e.g.  factor  (1), 

{x-2y){4:x-3y)=0. 
Hence  x  —  2y  =  0,4:X  —  3y  =  0. 

Combining  these   results  with  another  quadratic  equation 
produces  two  systems  of  the  preceding  kind. 

Ex.1.     Solve   I    J^-^-^^  +  ^r''  ^'^ 

\2i^-7xy-{-6y^  =  0.  (2) 

Factor  (2) ,  (x-2y){2  x  -  Sy)  =  0. 


SIMULTANEOUS  QUADRATIC  EQUATIONS 
Hence  we  have  to  solve  the  two  systems 


317 


From  (3) ,  x  =  2y. 

Substituting  in  (1), 

4y2_3y2  4.2j/  =  8, 
y^  +  2y-S  =  0, 
(y-l)(y  +  3)  =  0. 
Hence       y  =  1,  -»      ~  ^'  \ 


or 


<2x 

-3y  =  0, 

(8) 

-  3  ?/2  +  2  y 

=  8. 

(1) 

a; : 

=  fy. 

9f^ 
4 

-  8  y2  +  2  y 

=  3, 

3  2/2 

-Sy  +  12 

=  0, 

y='^^- 

^              6 

-80 

4+2V-5 

4-2 

V-5 

x  =  2+V-5, 


342.  If  both  equations  are  homogeneous  with  exception  of 
the  absolute  term,  the  problem  can  be  reduced  to  the  preceding 
case  by  eliminating  the  absolute  term. 


Ex.  2.    Solve 


3  a^  -  4  a;^  +  3  /  =  2, 
Eliminate  2  and  5  by  subtraction. 


(1)  x5, 

(2)  x2, 
Subtracting, 
Factoring, 
Hence  solve : 

(x-y  =  0, 

[2x^-2xy-\-5y^  =  5. 
From  (3),  x  —  y. 

Substituting  y  in  (2), 

2y2_2y2  4.5^,2=  6, 

2/2  =  1, 

y  =  ±i, 
x  =  ±l. 


15  x2  -2{)xy  +  15  ?/  =  2  X  5. 
4  a;2  -  4  x?/  +  10  2/2  =  5  X  2. 
11  a;2  -  16  x!/  +  5  2/2  =  0. 
(x-2/)(llx-5y)=0. 


(1) 
(2) 

(8) 
(4) 


or 


2x2 


llx-5|/  =  0, 

(3) 

2  x2  -  2  xy  -f  5  2^2  _  5. 

(2) 

y=:V», 

-  Y  a;2  +  1^1  x2  =  5, 

109  x2  _  g 

x  =  ±TbVl09,  |/=i^Vl09. 


318 


ELEMENTARY  ALGEBRA 


343.  In  general  two  quadratic  equations  which  are  homo- 
geneous with  the  exception  of  one  term  can  be  solved  if  these 
terms  are  similar.     For  the  similar  terms  can  be  eliminated. 


To  solve 


\2x'-2xy-4:y^-3y=:0. 

Eliminate  the  terms  containing  the  first  powers  of  y,  and 
proceed  as  before. 

EXERCISE  128 


1. 


Solve: 

[xy-^4:y^=S. 
x-^xy  =  6, 
7x'-\-2xy-5y''  =  0. 


*    \x'-4:xy-\-3y'  =  0. 

..{ 

e.[ 

'■( 
..   ( 

^      f3xy  +  f  =  7, 
[x^  -\-xy=:6. 

■    \2x'-4:xy-^Sy''=17. 


3a^-xy-2y^  =  0, 

-9xy-{-2f  =  -3. 
2x''-3xy-\-y'  =  0, 
x^  +  4.xy  +  2  /  =  17. 
2x^-xy-6y^  =  9, 
3x'-10xy-{-3y^  =  0. 
4:X^  =  9xy  —  5y^, 
7x^-3xy-3x-2y  =  l. 
a^  -\-6y^  =  5  xy, 
x^-{-2y^-3x-^2y  =  2. 


11. 


12. 


13. 


14. 


15. 


16. 


17. 


18. 


19. 


( x^  —  2xyz=5, 
[x'-y^  =  21. 


2x^-2xy  +  y^  =  2, 

X^-y^  =  -3. 

2x'-3xy-{-y''  =  3, 
x^-{-2xy-3y'  =  5, 

'3xy  +  y'  =  2S, 
.  4:  x^  -\- xy  =  8. 

^-xy-\-f  =  39, 
2x^-3xy  +  2y^  =  43. 

6x^-4.  xy~S  =  0, 
2xy-\-5y''-2S=^0, 

2a^-3xy-\-f  =  3, 


12x2- 


x^-^2xy-\-y^  =  9y, 

3xy-h2f  =  4y. 


'3x^-3xy  +  2y^  =  2x, 
.2  a^  ~\-3y^  —  4:X  =  xy. 


SIMULTANEOUS  QUADRATIC  EQUATIONS  319 


IV.    SPECIAL  DEVICES 

344.  Many  examples  belonging  to  the  preceding  types,  and 
others  not  belonging  to  them,  can  be  solved  by  special  devices. 

345-  ^.  Symmetrical  equations,  and  equations  which  would 
be  symmetrical  if  a  sign  were  changed,  can  often  be  solved  by 
the  substitution  x  =  u  +  Vf  y  =  u  —  v. 

Ex.1.   Solve  •     ^^  +  ''  =  ''  (^> 


,     fa^  +  /  =  2, 
\x-y  =  2. 


(2) 

Let  x=  u  +  V,  y  =  u  —  V. 
Substituting  in  (1),  (u  +  v)*  +  (w  —  v)*  =  2, 
or  ?t*  +  4  uH  +  6  7iH^  4-  4  uv^  +  v*, 

+  u*  -iuH  -\-6  uH"^  -  4  wi7»  +  V*  =  2, 


or                      2  w* 

+  12  wV                 +  2  V*  =  2. 

Dividing  by  2, 

M*  +  6  W^i;2  +  t;4  =  1. 

(3) 

Similarly  from  (2), 

2t>  =  2, 

v  =  L 

(4) 

Substituting  v  in  (3), 

w*  +  6  w2  +  1  =  1, 

or 

M*  +  6  w2  =  0. 

Factoring, 

m2(m2  +  6)  =  0. 

Therefore 

w  =  0,  w  =±V-6. 

Hence 

y 

=  ±V-6  +  ln +ln 
=  _tV-6-l.  J  -LJ 

Ex.    Solve 


346.  B.  Equations  of  higher  degree  can  sometimes  be  re- 
duced to  equations  of  the  second  degree  by  dividing  member  by- 
member. 

'0^  +  2/^  =  28,  (1) 

\x  +  y  =  4.,  (2) 

Dividing  (1)  by  (2),          x^-xy  +  y^  =  7.  (3) 

Squaring  (2),                   x^  ■]- 2  xy -^  y^  =  16.  (4) 
(4) -(3),                                        3xy  =  9, 

xy  =  3.  (5) 


320 

• 

ELEMENTARY  A 

XG^jE 

(3)- 

-(5). 

a;2- 

-  2  xy  +  2/2 
x-y 

=  4, 

=  ±2. 

Combining 

with 

(2), 

[y  = 

:  3,               r  X  : 

'    or  J 

:1.               [2/: 

=  1, 
=  3. 

347.  C.  Some  simultaneous  quadratics  can  be  solved  by  con- 
sidering not  X  OT  y  but  some  other  quantity,  as  -,  xy^  x^,  (aJ+2/)j 
x^y,  etc.,  at  first  as  the  unknown  quantity. 

Ex.  1.    Solve  ■!  , ^ 

[x  —  y  —  Vx  —  y  =  6.  (2) 


Considering  Vx  -\-  y  and  Vx  — «/  as  unknown  quantities  and  solving, 
we  have 

from  (1),  Vx  +  ?/  =  4  or  —  5, 

from  (2),  Vx  —  y  =  3  or  —  2. 

But  the  negative  roots  being  extraneous,  we  obtain  by  squaring, 
x  +  y  =  m, 
x~y  =  9. 
Therefore  x  =  12§,  y  =  3^, 

x^y'-\-xy-12  =  0,  (1) 


Ex.  2.    Solve    ,  ^  ..^ 

x-h?/-4.  (2) 

Solving  (1)  for  xy,  xy  =  3  or  —  4. 

r      xy  =  3,  r      XV  =  —  4, 

Combining  with  (2),  <!  '    or    i 

I  X  +  y  =  4.  I X  +  ?/  =  4. 

Solving  according  to  §  337,     x  -  ?/  =  2.  x  -  y  =  ±  4  V2. 

x  =  3,  ■|x=l,  1  x=2  +  2V2, 1  x=2-2\/2, 

2/  =  l-J2/  =  3.  J  2/=2-2V2.  J  ?/=2+2V2. 

Ex.3.    Solve  x^-\-4:y^-2x  +  4.y  =  27,  (1) 

a^i/  =  6.  (2) 

Multiplying  (2)  by  4  and  subtracting  from  (1), 

x2-4x?/  +  4y2-2x  +  4?/  =  3, 
or  (X  -  2  y)2  -  2  (X  -  2  «/)  -  3  =  0.  (8) 


SIMULTANEOUS  QUADRATIC  EQUATIONS 


321 


Considering  (x  -2y)  as  unknown  quantity  and  solving  (3), 

X  —  2  1/  =  3,  or  —  1. 
Hence  we  have  to  solve  tlie  two  systems 

(x-2y  =  -l,  jx-2y  =  3, 

I         xy  =  e.        L 
The  solution  produces  the  roots  : 


xy  =  6. 


tK  =  3, 

2/ =  2. 


+  3+V57  ]    +3-\/67 
■ 2 '     2 ' 


-  3  +  V57 
4 


-3-V57 


Ex.  1.   Solve  j 


348.  D.  If  the  quadratic  terms  can  be  eliminated,  the  example 
can  be  reduced  to  one  of  type  II  (one  equation  linear,  one 
quadratic). 

(2xy-x-\-2y  =  16,  (1) 

3xy-^2x-4:y  =  10.  (2) 

(1)  X  3,  6xy-Sx-\-6y  =  48.  (3) 

(2)  X  2,  6xy  +  ix-Sy  =  20.  (4) 

(4) -(3),  7  x-Uy  =  -28. 

Dividing  by  7,  x  —  2y  =  —  4. 

Hence,  x  =  2  y  —  4. 

Substituting  in  (I),  2  y(2  y  -  4)  -  (2  y  -  4)  +  2  y  =  16. 

Solving,  y  =  ^,]  -  1, 

r  or 
Substituting  in  (1),  x  =  2.  J  —  6. 


EXERCISE  129 
Solve  by  the  method  of  symmetrical  equations : 

x-y  =  l. 


1. 


raj*  +  2^  =  17, 
[x  +  y  =  S. 


3. 


2. 


a^  +  /  =  337, 
x-y  =  l. 


4. 


3^  +  2/^  =  82, 


322 


ELEMENTARY  ALGEBRA 


5. 


6. 


8. 


9. 


10. 


Solve  by  dividing  member  by  member : 

oc^  +  xy-\-y^  =  7, 

x-y  =  l. 

ic*  +  x2/ -4-2/^  =  3, 
ic^  —  ic?/  4-  2/^  =  !• 


(x'  +  f  =  12, 
I  ic  +  2/  =  6. 

x-y  =  l. 
x*-y*  =  SO, 
x'-y'  =  S. 


11 


Solve  by  the  method  of  §  347: 

(5x'  +  2y'  =  22, 


12. 


3x2-52/2  =  7. 
V2/2  +  6  =  5  xy, 


13. 


14. 


'x-{-y  =  25, 

.  Vx  +  V^  =  7. 


i^±y 


v^-^+\ 


X  "x  +  y 

ic2  +  2  2/2  =  76. 


4i, 


15 


Solve  by  eliminating  the  quadratic  terms : 

'  xy-\-x  =  15f  f  4  ic2  —  a;  +  2/  =  67, 

16.     1 
[xy-{-y  =  16.  [3x'-Sy  =  27. 

Ux'-5y'^-Sx-\-y  +  3  =  0, 
^^'     l8a^-102/'-7a;  +  92/  =  0. 


18. 


19. 


20. 


21. 


Solve  by  any  method : 
ra^  + 2/' =  1274, 
[x  =  5y. 

ja^H-2/«  =  16,021, 
{x-{-y  =  S7. 
r  x^  4-  xy  =  150, 
\y^^xy  =  75. 
fx2+a^-2/-  =  -31, 
\a^-'xy-^y^  =  4:9. 


22. 


24. 


25. 


a^  +  22/  =  8, 
a;2  +  2/2  =  13. 
r3a;  +  82/  =  20, 
\x'-^2y'-\-xy=5x-hy+l 
xy  +  3f  =  0, 
2/ =  26. 
a;  +  5  2/  =  3, 
a;2  +  3  a;?/  -  2  2/2  + 1  =  0. 


pa^-lOa- 
l3ar2_^  = 


SIMULTANEOUS   QUADRATIC  EQUATIONS  323 


26. 


27. 


28. 


29. 


30. 


31. 


32. 


33. 


34. 


35. 


36. 


37, 


38. 


5xy-9if  =  l, 
x^-2xy  =  0. 

x"^  —  y^  =  Q>5j 

x'-y'  =  5, 

x^-y"'  =  x  +  y, 
x-{-2y  =  7. 


(x'  +  xy  +  y'  =  3T, 
[x'  + 

(I 

(2x'-3y'~  =  6, 
[3x^-2y''  =  U 


+  2/' =  25. 

'■  +  3xy  +  2f'  =  20, 
x'  +  5y''-4:l  =  0. 


40  = 


19. 


f> 


1^  =  3. 
12/ 

ixy  =  12, 

1  2a; +  3?/ =  18. 

(5x'-\-y  =  3xy, 
\2x-y  =  0. 

5cc2_|_2/  =  22, 
3aj2-5/  =  7. 

x'-\-y'^  +  xy  =  UT, 
x-\-y  =  13. 
x2  +  2/2=130, 

^■hy 

x  —  y 


=  8. 


^4.x'  +  3xy-\-2f  =  lS, 
l3x''-\-2xy-y^  =  3. 


39. 


40. 


41. 


42. 


43. 


44. 


45. 


46. 


47. 


48. 


49. 


-2yX2x-3y)=26, 


((3x-2yX^ 
\x-j-l  =  2y. 


(x^-2xy-f  =  l, 
[3x'-4.xy  =  35. 
3x-  -\-  4xy  —  y^  =  14, 
2x'-{-5xy-^6y'-  =  9. 


I  x^ 


x-^y  -\-  -y/x  -\-y  =  12; 
x^  -\-y^  =  45. 
x''  +  y'  =  d7, 


x-^y  =  b. 


242. 


X- 

-y  = 

-2. 

2i 

v  —  y 

+  1 

3 

X 

-2y 

+  1 

8' 

XT 

-3xyA-f 

=  5. 

X 

y  = 

1:9, 

X 

6  = 

6:2/. 

1 

X 

5 
^6' 

1 

x" 

4 

_13 
36* 

x^ 

-? 

5 

1 

_1_ 

2/ 

1 

=  — . 

2 

1 

-J 

=  35, 

1 

X 

4-1. 

y 

=  5. 

324 


ELEMENTARY  ALGEBRA 


50. 


51. 


52. 


61. 


62. 


63. 


64, 


65. 


1-1  =  3. 

X     y 

1 2/(0.-^  +  /)  =  28. 


53. 


54. 


x-\/x  +  ?/ V^  _  35 
x^x  —  y^Jy      19 
a^  +  2/  =  13. 
:«  +  2/  =  520, 
-\/x  +  V^/  ~  -^^* 
1 


55 


a;      2/      3 


56 


57. 


58. 


r(a^  +  2/)(a^^  +  /)  =  175, 
\(»-2/)(cc'4-2/')  =  25.  [a^  +  2/2  =  160. 

{x^yJ  =  Zx^-2, 
{x-yY^?>f-\\. 
x'-2xy^^f-^(x-y)  =  0, 
2x^  +  xy-y'-9{x-y)=^0. 
'2x^-3xy  =  9(x-2y), 
.x'~3y'  =  6{x-2y). 

((x  +  2y)(x-]-Sy)  =  3(x-^y), 
'     [(2x-\-y)(3x  +  y)=2S(x  +  y). 
(x'-h2xy-hf-3  =  2(x  +  y\ 
[xy  =  2. 


I 


^x'+i/f  =  20, 
-\/x^  +  V^/  =  6. 

ar^ -f- 2/3  ^  9j^^ 

a;3  +  2/'  =  133, 

x^  —  xy  -\-y'^  =  19. 

^.2^.2/2  +  a.-  +  2/  =  18, 

a;2/  =  6. 

.^2  +  2/2  +  a^  —  2/  =  6, 

(.t2  +  2/2)(x-2/)  =  5. 


,,      r(a^+2/^)(a^+2/')=455, 
66.     i 

L  a?  +  2/  =  0. 


67. 


68. 


69. 


a?  4-  2/  =  m, 
a^  -f-  2/2  =  w. 


SIMULTANEOUS   QUADRATIC  EQUATIONS  325 


70.     {''-Tf  71. 

\.xy  —  y^  =  o^x. 


\(2a-xy-{2h-yy=a''-h\ 
(2a  —  x)(2b  —  y)  =  ah. 

x^—Sxy-\-^z'^=2t\,  {  x[x -\- y -\- z)  =  a, 

72.    j3a;  +  2/  =  5,  73.     |  ?/(ic  +  ?/ -f  2;)  =  6, 

.2x—Zz  =  —  l.  [z(x-{-y^z)  =  c. 

EXERCISE  130 
PROBLEMS 

1.  The  sum  of  two  numbers  is  44,  and  the  sum  of  their 
squares  is  1000.     Find  the  numbers. 

2.  The  difference  of  two  numbers  is  2,  and  the  sum  of  their 
squares  exceeds  their  product  by  103.     Find  the  numbers. 

3.    The  sum  ot  the  squares  of  two  numbers  is  40,  and  the 
product  of  the  numbers  is  equal  to  three  times  their  difference. 

4.  The  hypotenuse  of  a  right  triangle  is  13,  and  the  sum  of 
the  other  two  sides  is  17.     Find  these  sides.     (§  319.) 

5.  The  area  of  a  right  triangle  is  330  square  feet,  and  the 
hypotenuse  is  61  feet.     Find  the  other  two  sides. 

6.  The  hypotenuse  exceeds  one  side  of  a  right  triangle  by 
2  inches,  and  the  third  side  is  6  inches.  Find  the  unknown 
sides  of  the  triangle. 

7.  The  perimeter  of  a  right  triangle  is  70  feet,  and  its  area 
is  210  square  feet.     Find  the  three  sides. 

8.  The  mean  proportional  of  two  numbers  is  10,  and  the 
sum  of  their  squares  is  2504.     Find  the  numbers. 

9.  To  inclose  a  rectangular  field  68,200  square  feet  in  area, 
1460  feet  of  fence  are  required.  Find  the  dimensions  of  the 
field. 

10.   The  area  of  a  rectangle  is  1008  square  feet,  and  the  diag- 
onal (Ex.  23,  p.  303)  is  65  feet.     Find  the  length  of  the  sides. 


326  ELEMENTARY  ALGEBRA 

11.  The  sum  of  the  radii  of  two  circles  is  113  centimeters 
and  their  areas  are  together  equal  to  the  area  of  a  circle  whose 
radius  is  85  centimeters.      Find  the  radii.      (Area  of  circle 

12.  The  radii  of  two  spheres  differ  by  18  inches,  and  the 
difference  of  the  spherical  surfaces  is  equal  to  a  sphere  whose 
radius  is  48  inches.    Find  the  radii.   (Surface  of  sphere  =4 Tri^l) 

13.  A  dealer  sells  a  number  of  horses  for  $  1320,  receiving 
the  same  price  for  each  animal.  If  he  had  sold  one  horse  less, 
but  charged  $  10  apiece  more,  he  would  have  received  the  same 
sum.     Find  the  price  of  a  horse. 

14.  Two  cubes  together  contain  23|-  cubic  inches,  and  the 
edge  of  one,  increased  by  the  edge  of  the  other,  equals  4^  inches. 
Find  the  edge  of  each  cube. 

15.  The  volumes  of  two  cubes  differ  by  61  cubic  centimeters, 
and  the  edge  of  the  larger  exceeds  the  edge  of  the  smaller  by 
1  centimeter.     Find  their  edges. 

16.  A  number  less  than  100  is  equal  to  four  times  the  sum 
of  its  digits,  and  the  sum  of  the  squares  of  the  digits  is  20. 
Find  the  number. 

17.  The  sum  of  the  squares  of  two  numbers  added  to  the 
difference  of  the  numbers  equals  292.  The  same  sum  multi- 
plied by  the  difference  of  the  numbers  equals  3091.  What  are 
the  numbers  ? 


CHAPTER   XX 

PROPERTIES  OF  QUADRATIC  EQUATIONS 

CHARACTER  OF   THE  ROOTS 

349.    The  quadratic  equation  aoc^ -\- bx -\- c  =  0  has  two  roots, 


5+V&^-4ac  ^^^   -6-V6^-4ac         ^^322.) 


2a  2a 

Hence  it  follows : 

1.  If  b^—4:  ac  is  positive  or  equal  to  zero,  the  roots  are  real. 
If  h^  —  4:ac  is  negative,  the  roots  are  imaginary. 

2.  If  h^  —4,aG  is  a  perfect  square,  the  roots  are  rational. 

If  b^  —  4:ac  is  not  a  perfect  square,  the  roots  are  irrational. 

3.  If  b^  —  4:ac  is  zero,  the  roots  are  equal. 

If  6^  —  4  ac  is  not  zero,  the  roots  are  unequal. 

350.    The  expression  6^  — 4ac  is  called  the  discriminant  of 

the  equation  ax^  +  6x  +  c  =  0. 

Ex.  1.    Determine  the  character  of  the  roots  of  the  equation 
3x'-2x-5  =  0. 

The  discriminant  =  ( -  2)2  -  4 . 3  •  ( -  5)  =  64. 
Hence  the  roots  are  real,  rational,  and  unequal. 

Ex.  2.   Determine  the  character  of  the  roots  of  the  equation 
4ic2-12a;  +  9  =  0. 

Since  (—  12)2  —  4.4.9  =  0,  the  roots  are  real,  rational,  and  equal. 

Ex.  3.    Prove  that  the  roots  of  the  equation   x^-i-2px+p^ 
—  q^  —  2qr  —  r^  =  0  are  rational. 

The  discriminant  =  (2p)2 -i(p2- q2-2  qr- r^), 
=  4(52  +  2gr  +  r2),. 
=  4(5  +  02. 

Hence  the  roots  are  rational. 

327 


328  ELEMENTARY  ALGEBRA 

351.  The  preceding  propositions  make  it  possible  to  deter^ 
mine  the  coefficients  so  that  the  roots  shall  satisfy  a  given 
condition. 

Ex.  1.    Determine  the  value  of  m  for  which  the  roots  of  the 
equation  —  ^a^ -{-^x-\-3  =  7n  are  equal. 
Transposing,  —  |  ic^  +  |  x  +  (3  —  m)  =  0. 

The  discriminant  must  equal  zero.     (§  349.) 
Hence  (l)"^  -  4  (-  i)  (3  -  m)  =  0. 

Or  I  +  3  -  m  =  0. 

Hence  m  =  6^. 

Note.  This  result  can  be  obtained  by  inspection  of  the  graph  of  this 
function,  which  was  discussed  in  §302. 

Ex.  2.  Determine  the  value  of  m  for  which  the  equation 
(m  -\-5)x^-\-3  mx  —  4  (m  —  5)  =  0  has  equal  roots. 

The  discriminant  must  equal  zero. 

Hence  (3  m)^  +  4  . 4  (m  +  5)  (m  -  5)  =  0. 

Or'  25  to2  -  400  =  0. 

Therefore  wi  =  +  4,  or  —  4. 

Check.  The  equations  9  x^  +  12  a;  +  4  =  0,  and  x^  -  12  a:  +  36  =  0,  have 
equal  roots. 

EXERCISE  131 

Determine,  without  solution,  the  character  of  the  roots  of 
each  equation : 

1.  x'-lx-\-12  =  0.  10.  ^=\x-l. 

2.  3a}2-10a5  +  3  =  0.  11.  a;^  +  15 a;  =  17. 

3.  9a^-6a;  +  l  =  0.  12.  9 x^  =. 24 a; -  16. 

4.  jr2_^  2  a; +  9723  =  0.  13.  9  a;^  =  17  ic  -  47,125. 

5.  a^  — 7aj— 7  =  0.  4      17 

6.  2x'-^x  +  l  =  0.  ^*-  ^^+^  =  ^' 

7.  6a^-5a;-l  =  0.  ^^     a;-8 

15.    =  x. 

8.  a^  =  a;4-12.  x-3 

9.  a^-a;V2  =  3.  16.    (a;  +  4) (a?  + 13)  =  90. 


PROPEBTIES   OF  QUADRATIC  EQUATIONS         329 

Determine  the  value  of  m  for  which  the  roots  of  the  follow- 
ing equations  are  equal : 

17.  2x^-8x-\-7n  =  0.  22.  Ax^-2x  =  Sm. 

18.  3a^  +  6x  +  m  =  0.  23.  x^-6x  +  m^0. 

19.  5x^  +  20x  =  m.  24.  a^- (m  +  3)a; +  3m  =  0. 

20.  ma^  — 14  aj  — 7  =  0.  25.  a^+ (m  +  5)x-{-5m  =  0. 

21.  x^-{-2x  +  3  =  m.  26.  (m  +  l)a^  +  3ma;  =  l  —  m. 

27.  ic2_(2m-3)a;4-2m  =  0. 

28.  (13  +  3m)a^  +  5ma;  +  13-3m  =  0. 

RELATION   BETWEEN   ROOTS  AND  COEFFICIENTS 

352.   If  the  roots  of  the  equation  ax- -^  bx -{- c  =  0  are  denoted 


by  ri  and  r2,  then  _5  4.V6^34 


2a 


ac 
— ) 


_5_V62-4ac 

2a 

b 

r,  +  r,  =  --, 

Hence 

and 

,^,^,(-^)--(^;-4ac). 

Or 

c 

h        c 

If  the  given  equation  is  written  in  the  form  a^  +  -aj  +  -  =  0, 

these  results  may  be  expressed  as  follows : 

353.   If  the  coefficient  of  x^  in  a  quadratic  equation  is  unity, 
(a)    The  sum  of  the  roots  is  equal  to  the  coefficient  ofx  with  the 
sign  changed. 

(6)    The  product  of  the  roots  is  equal  to  the  absolute  term. 

E.  g.  the  sum  of  the  roots  of  4  x^  +  5  x  —  3  =  0  is  —  |,  their  product 
is  —  |. 


330  ELEMENTARY  ALGEBRA 


> 


354.    Formation  of  equations.     If  r^  and  rg  denote  the  roots  of 

b         c 
the  quadratic  equation  x^  -{--x-{--  =  0,  the  equation  may  be 

written : 

c(^-{n  +  n)x  +  r^r.,  =  0.  (1) 

Or  factoring,  (x  —  r^)  (x  —  rg)  =  0.  (2) 

To  form  an  equation  whose  roots  are  ^iven  we  may  use 
either  (1)  or  (2). 

Ex.  1.    Form  the  equation  whose  roots  are  2  and  —3. 
According  to  (2),  (x  -  2)  (a;  +  3)  =  0. 

Or  x'^  +  x-6  =  0. 

Ex.  2.    Form  the  equation  whose  roots  are  —  f  and  —J. 
The  sum  of  the  roots  =  —  4. 

The  product  of  the  roots  =  +  -^^. 

Hence,  according  to  (1),     x"^  +  4:X  +  ^-  =  0. 
Multiplying  by  4,  4:X^  +  16x+15=  0. 

Ex.  3.   Form   the   equation  whose  roots   are   2+V2  and 
2-V2. 

The  sum  of  the  roots  =  4. 

The  product  of  the  roots  =  2. 

Hence  the  equation  is  x^  —  4:X  +  2  =0. 

EXERCISE   132 

In  each  of  the  following  equations  determine  by  inspection 
the  sum  and  the  product  of  the  roots: 

1.  x'-7x  +  6  =  0.  4.   5x'-\-5x-\-l  =  0. 

2.  a;2  +  8aj-2  =  0.  5.    x^- (a  +  b)x  +  ab  =  0. 

3.  8x^-{-5x-\-3  =  0.  6.    7a;2-.aj  +  l  =  0. 


PROPERTIES   OF  QUADRATIC  EQUATIONS         331 


Form  the  equations  whose  roots  are 

: 

7.    1,   2. 

14. 

-h  -f. 

8.    -5,  -6. 

15. 

3  4-V2,  3- 

-V2. 

9.    -1,  +6. 

16. 

a  -f-  V6,   a  - 

-V6. 

10.  —a,  +a, 

11.  i    3. 

17. 

14-V3     1 
2      ' 

-V3 

2 

12.  4  +  V3,   4- 

13.  ab,   a. 

-V3. 

18. 

a  +  V-6 
4 

a-V-ft 
4 

Solve  the  following  equations,  and  check  the  answers  by 
forming  the  sum  and  the  product  of  the  roots: 

19.  a^  — 4»  +  l  =  0.  21.    a^-6aj  +  4  =  0. 

20.  a^-6x-\-6  =  0.  22.    0.-2  +  0;  + 1  =0. 

23.  Without  solving  find  the  sum  of  the  squares  of  the  roots 
of  the  equation  aoc^  -\-  bx  -\-  c  =  0. 

24.  Without  solving  find  the  difference  of  the  roots  of  the 
equation  aa^  +  6a;  +  c  =  0. 

FACTORING  OF   QUADRATIC   EXPRESSION'S 

355.  Let  ri  and  rg  denote  the  roots  of  the  equations 

ax^ -{■  bx  +  c  =  0. 

aa^ 4-  bx -{-  c  =  al x^  +  -X  +  -] 
\        a       aj 

=  a  (ar'  -  [ri  +  r^-\  x  +  r^r^),     (§  354.) 

Or  factoring,     ajr^ -\-  bx  +  c  =  a{x  —  r-^{x  —  r^. 

356.  Hence  any  quadratic  expression  can  be  factored.  TJie 
factors,  however,  are  rational  only  if  the  roots  of  the  equation 
obtained  by  making  the  exjjression  equal  to  zero  are  rational. 


332  ELEMENTARY  ALGEBRA 

Ex.  1.    Deterinine  whether  6a:r  +  9x-{-2  has  rational  factors. 

The  discriminant  =  92  -  4  •  6  •  2  =  33. 

Hence  the  roots  are  irrational,  and  the  expression  has  no  rational 
factors. 

^    Ex.2.    Eactor  3aj2_;1^9^._j^4 

Solving  the  equation  3  a:^  —  19  x  —  14  =  0  by  the  formula, 


_  19  ±  V19^  +  4  .  3  .  14 

Or  ^^11±23^       ^^_2. 

6  '  3 

Hence  3x^  -  19x  -  14  =  3(x  +  f)(a;  -  7) 

=  (3aj  +  2)(ic-7). 

Ex.  3.    Eactor  2ay^-2x-{-l. 

Solving  the  equation  by  means  of  the  formula,  we  find  the  roots 

2 


Hence 


2x^-2x  +  l  =  2{x-.l±^^{x-l^^:^^ 


=  2 

=  i(2x  -  1  ~y/'^)(2x  -  1  +V^^). 

Ex.4.    Eactor  x^ -{-xy -2xy^ -2y^ -{-Sf -Sy\ 

The  expression  is  quadratic  in  respect  to  x,  hence  we  solve  the  equation 

^2  +  x(y  -  2 ?/2) -  2 2/2  +  8 2/3  _  8?/4  =  0. 
By  formula,  x  =  —  2  y  +  4  y'^,  ot  y  —  2  y'^. 

Hence        x:^  +  xy  -  2 xij^  -  2y^  -\-  S y^  -  S y^ 

=  (x  +  2y  -  iy^)(x  -  y  +  2y^). 

Note.  A  quadratic  equation  cannot  have  three  roots.  For  if  we 
write  the  equation  ax^  +  &x  +  c  =  0  in  the  form  a(x  —  ri)(x  —  r^)  =  0,  no 
other  value  r^,  not  equal  to  either  ri  or  ra,  can  satisfy  the  equation,  as 
«(»'a  —  ri)(r3  —  ^2)  cannot  equal  zero. 


PROPERTIES   OF  QUADRATIC  EQUATIONS         333 


EXERCISE   133 


Determine  whether  the  following  expressions  have  rational 
factors : 


1. 

Zx'-lx  +  n. 

6. 

9m2-6mn-8w2. 

2. 

6a^  +  20x-65. 

7. 

16  0^  +  101  a; -125. 

3. 

72  0^-145  a; +  72. 

8. 

15a^-19x?/-562/'. 

4. 

a;2_82a;  +  1572. 

9. 

4a^-20  6x+2562_9a2. 

5. 

m^  —  mn  —  2  v?. 

10. 

(A;  +  l)x2  +  3A:a;  +  A;-l. 

Eesolve  into  factors : 

11.  ic2-60aj  +  899.  17.  300  aj^  +  811  a;  -  630. 

12.  25  0^2  _  100  a? +  96.  18.  a;^  ^  214  a;  +  11,448. 

13.  3.-2 -90  a; +  2009.  19.  a^  +  aj  +  1. 

14.  50 a^  + 315 a; -729.  20.  a;2  +  3a;-3.' 

15.  a;2-64a;  +  1015.  21.  2ar'  +  a;-2. 

16.  AQ3?-b3xy-\-Qy\  22.  aj^  +  l. 

23.  aa;2  +  (a^  +  l)a;  +  a. 

24.  ^x^-4.xy  +  Sxz-4:y^-\-%yz-Zz\ 

25.  1)2 +  7^5  +  2^2. 

26.  d>:x?  -lQ>xy-^xz  +  Qy'^-Syz  —  ZOz\ 

27.  12y-18pg  +  28i)r-12g2-fl9g7._5y8 

28.  a:2_3a;_3. 


CHAPTER  XXI 
PROGRESSIONS 

357.  A  series  is  a  succession  of  numbers  formed  according 
to  some  fixed  law. 

The  terms  of  a  series  are  its  successive  numbers. 

ARITHMETIC  PROGRESSION 

358.  An  arithmetic  progression  (A.  P.)  is  a  series,  each  term 
of  which,  except  the  first,  is  derived  from  the  preceding  by 
the  addition  of  a  constant  number. 

The  common  difference  is  the  number  which  added  to  each 
term  produces  the  next  term. 

Ttius  each  of  the  following  series  is  an  A.  P. : 
3,  7,  11,  15,  19,  .... 
17,  10,  3,   -4,  -11,  .... 
a,  a  +  d,  a-\-2d,  a  +  3d,  •••. 

The  common  differences  are  respectively  4,  —  7,  and  d. 

The  first  is  an  ascending,  the  second  a  descending,  progression. 

359.  To  find  the  nth  term  /  of  an  A.  P.,  the  first  term  a  and 
the  common  difference  d  being  given. 

The  progression  is  a,  a  +  d,  a  +  2  d,  a-\-3  d. 

Since  d  is  added  to  each  term  to  obtain  the  next  one, 

2  d  must  be  added  to  a,  to  produce  the  3d  term, 

3  d  must  be  added  to  a,  to  produce  the  4th  term, 

(?i  —  l)d  must  be  added  to  a,  to  produce  the  wth  term. 

Hence  l  =  a-\-{n-l)d.  (I) 

Thus  the  12th  term  of  the  series  9,  12,  15  is  9  +  H  •  3  or  42. 

334 


PROGRESSIONS  335 

360.  To  find  the  sum  5  of  the  first  n  terms  of  an  A.P.,  the  first 
term  a,  the  last  term  /,  and  the  common  difference  d  being  given. 

s  =  a  +  (a  +  d)  +  (a  +  2d)  •••  (l  —  d)-{-l. 
Ee versing  the  order, 

s  =  l+(l-d)  +  (l-2d)"'(a-^d)+a. 
Adding,     2  s  =  (a  -\- 1)  +  {a  + 1)  +  (a  -\- 1)  -•-  (a  + 1)  +  (a  + 1). 
Or  2s  =  n(a  +  Z). 

Hence  s  =  ^(a  +  0.  (II) 

jj 

Thus  to  find  the  sum  of  the  first  50  odd  numbers,  1,  3,  5  •••  we  have 
from  (I),  Z  =  1  +49.2  =99. 

Hence  s  =  -^f  (1  +  99)  =  2500. 

361.  In  most  problems  relating  to  A.  P.,  jive  quantities  are 
invoiced  ;  hence  if  ayiy  three  of  them  are  given,  the  other  two  may 
he  found  by  the  solution  of  the  simultaneous  equations  : 

(/=  a +(/»-!)</.  (I) 

«  =  |(a  +  /).  (II) 

Note.  It  is  possible  to  find  general  formulae  expressing  any  two 
quantities  in  terms  of  any  three  others.  The  formulae,  however,  have 
httle  value,  since  all  examples  can  be  solved  without  them. 

Ex,  1.  The  first  term  of  an  A.  P.  is  12,  the  last  term  144,  and 
the  sum  of  all  terms  1014.     Find  the  serieso 

s  =  1014,  a  =  12,1  =  144. 
Substituting  in  (I)  and  (II), 

144  =  12  +  (n  -  1)  d.  (1) 

1014  =  ^(12  +  144).  (2) 

From  (2),  78  n  =  1014,  or  =  n  =  13. 

Substituting  in  (1),        144  =  12  +  12  •  d. 

Hence  d  =  11. 

The  series  is,  12,  23,  34,  45,  56,  67,  78,  89,  100,  111,  122,  133,  144. 


336  ELEMENTARY   ALGEBRA 

Ex.  2.    Find  n,  if  s  =  204,  d  =  Q,l  =  49. 


Substituting, 

49  =r  a  +  (n  -  1)  •  6. 

(1) 

204  =  -  (a  +  49). 

(2) 

From  (1), 

a  =  49 -(71 -1)6. 

Substituting  in  (2), 

204  =  ^(98-^-1.6). 

408  =  w(104-6w). 

6  m2  -  104  w  +  408  =  0. 

3  n2  -  52  n  +  204  =  0. 

Solving, 

w  =  6,  or  llf 

But  evidently  n  cannot  be  fractional,  hence  n  =  6. 

EXERCISE   134 

1.  Find  the  11th  term  of  the  series  11,  22,  33,  .... 

2.  Find  the  18th  term  of  the  series  94,  87,  80,  ••.. 

3.  Find  the  7th  term  of  the  series  12,  151  19. 

4.  Find  the  15th  term  of  the  series  8,  lOJ,  12f 

5.  Find  the  11th  term  of  the  series  —1,  —  3i,  —6. 

6.  Find  the  12th  term  of  the  series  —  7,  —  1,  +5. 

7.  Find  the  13th  term  of  the  series  -  8.5,  -  10.9,  -  13.3c 

Find  the  last  term  and  the  sum  of  the  following  series  ; 

8.  4,  7,  10,  ...,  to  11  terms. 

9.  —  2,  —  1,  0,  ...,  to  14  terms. 
10.  4,  —1,  —6,  ...,  to  13  terms, 
11-    if,|, -,  tolOterms. 

Find  the  sums  of  the  following  series : 

12.  a?  — 5,  a;  — 4,  a;  — 3,  •••,  to  11  terms, 

13.  -1  .4,  -1  ^3,  -1  .2,  •..,  toll  terms. 


J^ROGRESSIONS  337 

14.  11  3i,.5,  •••,  to  5  terms. 

15.  2 X  -^  3 y,  3  X  -{-  2 y,  A X  -{-  y,  •••,  to  7  terms. 

16.  Find  the  sum  of  the  first  25  integral  numbers. 

17.  Prove  that  the  sum  of  the  first  n  integral  numbers  is 
n(n-{-l) 

2 

18.  Find  the  sum  of  the  first  49  odd  numbers. 

19.  Find  the  sum  of  the  first  n  odd  numbers. 

20.  Find  the  sum  of  the  first  40  even  numbers. 

21.  The  first  term  of  an  A.  P.  is  13,  the  last  term  88,  and 
the  common  difference  is  5.     Find  the  number  of  terms. 

22.  The  first  term  of  an  A. P.  is  62,  the  last  term  is  7,  and 
the  common  difference  is  —  5.     Find  the  number  of  terms. 

23.  The  last  term  of  an  A.  P.   of  22  terms  is  5,  and  the 
common  difference  is  ^.     Find  the  first  term  and  the  sum. 

24.  Given  a  =  4,  1  =  25,  n=S.     Find  d. 

25.  Given  a  =  7,  fZ  =  4,  n  =  43.     Find  I  and  s, 

26.  Given  a  =  —  3,  d  =  2,  n  =  8.     Find  I  and  s. 

27.  Given  a  =  4,  n  =  12,  Z  =  26.     Find  d  and  s. 

28.  Given  a  =  5,  w  =  4,, Z  =  —  2.     Fi;id  d  and  s. 

29.  Given  a  =  1,  n  =  35,  s  =  1225.     Find  d  and  I 

30.  Given  n  =  12,  cZ  =  4,  s  =  99.     Find  a  and  Z. 

31 .  Given  d  =  1|,  n  =  33,  I  =  77.     Find  a  and  s. 

32.  Given  a  =  21,  Z  =  -  59,  s  =  -  323.     Find  (Z  and  w 

33.  Given  Z  =  23,  s  =  58,  n  =  29.     Find  a  and  d. 

34.  Given  a  =  —  7,  (Z  =  3,  s  =  430.     Find  li  and  Z. 

35.  Given  Z  =  97,  cZ  =  3,  s  =  1612.     Find  a  and  n, 

36.  Find  d  in  terms  of  a,  n,  and  Z. 

37.  Find  n  in  terms  of  a,  I,  and  s. 


338  ELEMENTARY  ALGEBRA 

362.  When  three  numbers  are  in  A.  P.  the  second  one  is 
called  the  arithmetic  mean  between  the  other  two. 

Thus  X  is  the  arithmetic  mean  between  a  and  b,  if  a,  x,  and 
6  form  an  A. P.,  or  if        ^_^_^_^, 

a  1   •  a  +  6 

Solving,  x  =  —^- 

I.e.  the  arithmetical  mean  hetweeyi  tico  numbers  is  equal  to 
half  their  sum. 

Ex.  1.    Insert  5  arithmetic  means  between  14  and  62. 

The  number  of  terms  is  evidently  7. 
Hence  n  =  7,  «  =  14,  Z  =  62. 

Substituting  in  (I),  62  i=  14  +  6  •  d. 

Solving,  d  =  S. 

Therefore  the  series  is  14,  22,  30,  38,  46,  54,  62. 
Or  the  means  are  22,  30,  38,  46,  54. 

Ex.  2.  The  eleventh  term  of  an  A.  P.  is  39,  the  nineteenth 
term  is  67.     Find  the  series. 

Using  formula  (I)  twice,  o9  =  a  +  10  •  d.  (1) 

67  =  «  +  18  d.  (2) 

Subtracting  (1)  from  (2),        28  =  8  d 
Whence  cZ  =  |. 

Substituting  in  (1),  39  =  a  +  35. 

Therefore  a  =  4. 

Or  the  series  is  4,  7^,  11,  14^,  •.-. 

EXERCISE  135 

1.  Insert  6  arithmetic  means  between  25  and  4. 

2.  Insert  4  arithmetic  means  between  17  and  — 13. 

3.  Insert  5  arithmetic  means  between  —  1  and  —  7. 

4.  Insert  2  arithmetic  means  between  —  9  and  12. 

5.  Insert  9  arithmetic  means  between  2^  and  27|-. 


PBOGBESSIONS  339 

Find  the  arithmetic  means  between : 

6.  a  and  5  a.  8  ^   and  — tX. 

11  '  X  X 

7.  -and—  _     a  —  b       ,3a  +  & 
a         6                                 9.   -Y-  ^^^  ~2      ° 

Find  the  series  in  which : 

10.  The  5th  term  is  19,  and  the  8th  term  is  31. 

11.  The  9th  term  is  12,  and  the  17th  term  is  60. 

12.  The  4th  term  is  —  1,  and  the  9th  term  is  —  2. 

13.  The  12th  term  of  an  A.  P.  is  14,  and  the  20th  term  is 
—  6.     Find  the  28th  term. 

14.  The  sum  of  three  numbers  in  A.  P.  is  18,  and  their 
product  is  192.     Find  the  numbers. 

Hint.  Let  x  —  y,  x,  x  +  y,  represent  the  numbers. 

15.  The  sum  of  three  numbers  in  A.  P.  is  9,  and  the  sum  of 
their  squares  is  29.     What  are  the  numbers  ? 

16.  The  sum  of  five  numbers  in  A.  P.  is  25,  and  the  sum  of 
their  squares  is  135.     Find  the  numbers. 

17.  The  sum  of  four  numbers  in  A.  P.  is  20,  and  their  product 
is  384.     Find  the  numbers. 

Hint.    Let  x  —  Sy,  x—  y,  x  +  y,  x-]-Sy  represent  the  A.  P. 

18.  How  many  times  does  a  common  clock  strike  in  12  hours? 

19.  A  bookkeeper  receives  every  year  an  increase  of  $  100. 
How  many  dollars  did  he  receive  in  15  years  if  he  received 
$  1200  during  the  first  year  ? 

20.  For  boring  a  well  200  yards  deep  a  contractor  receives 
$1.00  for  the  first  yard,  and  for  each  yard  thereafter  1^  more 
than  for  the  preceding  one.  How  much  does  he  receive  all 
together  ? 


340  ELEMENTARY   ALGEBBA 

21.  A  man  saved  each  month  $  1.00  more  than  in  the  preced- 
ing one,  and  all  his  savings  in  10  years  amounted  to  f  8460. 
How  much  did  he  save  the  first  month  ?     How  much  the  last  ? 

22.  $  2000  is  divided  among  eight  persons  so  that  each  per- 
son receives  $  10  more  than  the  preceding  one.  How  much 
does  each  receive  ? 

23.  A  man  accepts  a  position  at  a  salary  of  ^1200  for  the 
first  year,  and  is  to  receive  every  year  ^  50  increase  of  salary- 
In  how  many  years  will  he  have  received  ^45,000? 

24.  If  a  body  falls  5  meters  during  the  first  second,  3  times 
as  far  during  the  next  second,  5  times  as  far  during  the  next 
second,  etc.,  how  far  will  it  fall  during  the  7th  second  ?  during 
the  ?ith  second? 

25.  How  far  will  a  body  fall  in  7  seconds  ?   in  t  seconds  ? 

26.  A  man  pays  off  a  debt  of  $24,000  by  monthly  payments 
of  $100,  and  the  first  payment  is  made  at  the  end  of  the  first 
month.  If  the  yearly  rate  of  interest  is  6%,  what  is  the  total 
amount  of  interest  paid  until  the  debt  is  cleared  off  ? 

27.  A  man  purchases  a  $500  piano  by  paying  monthly 
installments  of  $10  and  interest  on  the  debt.  If  the  yearly 
rate  of  interest  is  6%,  what  is  the  total  amount -of  interest? 

GEOMETRIC  PROGRESSION 

363.  A  geometric  progression  (G.  P.)  is  a  series  each  term  of 
which,  except  the  first,  is  derived  from  the  preceding  one  by 
multiplying  it  by  a  constant  number,  called  the  ratio. 

E.g.  4,  12,  36,  108,  .... 
4,  -2,  +1,  -i,  .... 
a,  ar,  ar^,  ar^,  .... 
The  ratios  are  respectively  3,  —  |,  and  r. 

364.  To  find  the  nth  term  /of  a  G.  P. ;  the  first  term  a  and 
the  ratios  r  being  given. 

The  progression  is  a,  ar,  ar^j  •••. 


PROGRESSIONS  341 

To  obtain  the  71th.  term  a  must  evidently  be  multiplied  by 

Hence  Z  =  a?-"-\  (I) 

Thus  the  5th  term  of  the  series  16,  24,  36,  ••.,  is  16(f)*  or  81. 

365.  To  find  the  sum  5  of  the  first  n  terms  of  a  G.  P.,  the  first 
term  a  and  the  ratio  r  being  given. 

s  =  a -\- ar  -{-  ar^  ••  •  ar'^'K  (1) 

Multiplying  by  r,  rs=        ar -{-ar^  •'•  -\-  a?-".  (2) 

Subtracting  (1)  from  (2), 

s  (r  —  1)  =  ar"—  a. 

Therefore  g  =  ^^'"~/-  (II) 

r  —  1 

Thus  the  sum  of  the  first  6  terms  of  the  series  16,  24,  36,  •••. 

s  =  16  [(1)^-1]  ^  32  (7^JL  -  1)  =  332|. 

Note.     If  w  is  less  than  unity,  it  is  convenient  to  write  formula  (II)  in 

the  following  form :  a-ar^  .tttn 

s  = (III) 

1  —r 

366.  In  most  problems  relating  to  G.  P.  Jive  quantities  are  in- 
volved;  hence,  if  any  three  of  them  are  given,  the  other  two  may 
be  found  by  the  solution  of  the  simultaneous  equations : 

n  =  ar"-\  (I) 

Ex.  1.   To  insert  5  geometric  means  between  9  and  576. 

Evidently  the  total  number  of  terms  is  6  +  2  or  7. 
Hence  n  =  7,  a  =  9,  I  =  576. 
Substituting  in  I,  576  =  9  »*. 

r^  =  64. 
r=±2. 
Hence  the  series  is  9,  18,  36,  72,  144,  288,  576. 

or  9,  -  18,  36,  -  72,  144,  -  288,  576. 

And  the  required  means  are  ±18,  3,  ±762,  144,  ±288. 


342  ELEMENTARY  ALGEBRA 

EXERCISE  136 

1.  Find  the  8th  terra  of  the  series  5,  10,  20, .... 

2.  Find  the  8th  term  of  the  series  2,  6,  18,  •••. 

3.  Find  the  7th  term  of  the  series  3,  —  6,  +  12, .... 

4.  Find  the  6th  term  of  the  series  4,  —  6,  +  9,  •••. 

5.  Find  the  10th  term  of  the  series  81,  27,  9,  •••. 

6.  Find  the  7th  term  of  the  series  —4,  +  f ,  —  i^-, ..«. 

Find  the  sum  of  the  following  series : 

7.  2,  6,  18,  •••  to  6  terms. 

8.  2,  —  4,  8,  •••  to  7  terms. 

9.  1,  1,  i  •••to  8  terms. 

10.  27,  18,  12,  •••to  6  terms. 

11.  48,  36,  27,  •••to  5  terms. 

12.  729,  -  243,  81,  •••  to  5  terms. 

13.  a^,  a^,  aV"  to  8  terms. 

14.  aP  +  a%  +  a^b^  •  •  •  to  7  terms. 

15 .  a^  —  a^b-{-  a^b^,  •  •  •  to  7  terms. 

16.  Find  the  geometric  mean  between  2  and  50. 

17.  Prove  that  the  geometric  mean  of  two  numbers  is  equal 
to  the  mean  proportional  between  the  numbers. 

18.  Find  the  geometric  mean  between  a-  —  b^  and  -^ — 

a  —  b 

19.  Insert  3  geometric  means  between  4  and  324. 

20.  Insert  3  geometric  means  between  3  and  48. 

21.  Given  r  =  4,  n  =  3,  Z  =  80,  find  a  and  s. 

22.  Given  r  =  3,  n  =  3,  1  =  1S,  find  a  and  s. 

23.  Given  ?-  =  4,  n  =  3,  s  =  105,  find  a  and  I. 

24.  Given  r  =  5,  ii  =  4,  s  =  780,  find  a  and  I. 


PROGRESSIONS  343 

25.  Given  a  =  2,  r  =  4:,  1  =  32,  find  n  and  s. 

26.  Given  a  =  5,  ?•  =  4,  1=  80,  find  n  and  s. 

27.  Given  a  =  4,  7i  =  3,  Z  =  64,  find  r  and  s. 

28.  Given  a  =  5,  n  =  3,  Z  =  125,  find  r  and  s. 

29.  Given  a=15,  r  =  3,  s  =  600,  find  n  and  I 

30.  Given  a  =  15,  r  =  4,  s  =  5115,  find  71  and  Z. 

31.  Find  s  in  terms  of  a,  r,  and  Z. 

32.  Find  a  in  terms  of  ?•,  n,  and  Z. 

33.  Find  a  in  terms  of  r,  n,  and  s. 

34.  Find  7'  in  terms  of  a,  n,  and  I. 

35.  Find  the  sum  of  the  series  V3,  3,  3  a/3,  •••  to  5  terms. 

36.  Find  the  sum  of  the  series  1,  2,  4,  8,  to  n  terms. 

37.  The  fourth  term  of  a  G.  P.  is  135,  the  seventh  terra 
3645.     Find  the  series. 

38.  The  sum  of  the  third  and  fifth  terms  of  a  G.  P.  is  90, 
and  the  sum  of  the  sixth  and  eighth  terms  is  2430.  Find  the 
series. 

39.  The  population  of  a  city  is  100,000,  and  it  increases 
50%  every  4  years.     What  will  the  population  be  in  20  years  ? 

40.  A  sum  of  money  invested  at  6%  compound  interest 
doubles  itself  in  12  years.  What  will  $1.00  invested  at  6% 
compound  interest  amount  to  in  240  years  ? 

INFINITE   GEOMETRIC  PROGRESSION 

367.  If  the  value  of  r  of  a  G.  P.  is  less  than  unity,  the  value 
of  r"  decreases,  if  n  increases.  The  formula  for  the  sum  may 
be  written  ^  _  ^^„         ^  ^^,„ 

s 


1—r       1—r     1 

ntly  large,  r",  a: 
made  less  than  any  assignable  number. 


By  taking  n  sufficiently  large,  r",  and  hence  ,  may  be 


344 


ELEMENTARY  ALGEBRA 


Consequently,  by  taking  a  sufficiently  large  number  of  terms, 

—  by  less  than  any  given  num- 


s  can  be  made  to  differ  from 


ber,  however  small.     This  is  usually  expressed  by  the  formula 

1  —  r 

rvhere  s^  denotes  the  "  sum  to  infinity." 

Ex.  1.    Find  the  sum  to  infinity  of  the  series  1,  —  ^,  i,  ••- 


Therefore 


a  =  1,  r  = 

1 


Tt-r^- 


Ex.   2.    Find  the  value  of  .3727272  .... 

.3727272  ...  =  .3  +  .072  +  .00072  + 
The  terms  after  the  first  form  an  infinite  G.  P. 


Hence 


a  =  .072,  r  =  .01. 

^    .072     ^  .072  ^  72  ^  4 

°°      l-.Ol       .99       990      55" 

41 


Therefore      .37272  . . .  =  A  +  A 

10     55      110 


EXERCISE  137 
Find  s^  for  each  of  the  following  series 


1.    1,1  i-..       3.    2,  -f,  + 


5?    -^2-5 


o      1     1     1 


4.    4,1,  J..., 

Find  the  value  of : 
6.   .777  ....  8.    .126126  •- 


7.    .545454  ....    9.    .42727 

L2.    The  sum  of 
4.     Find  the  series 


5.    3,  -i,+^ 


10.  .42111  .... 

11.  3,  V3,  1  • 


12.    The  sum  of  .an  infinite  G.  P.  is  6,  and  the  first  term  is 


PROGEESSIONS  345 

13.  A  ball  is  thrown  vertically  upwards  to  a  height  of  16 
feet.  After  striking  the  ground  it  rebounds  to  three  fourths 
the  height  it  dropped  from,  and  so  on  for  each  successive  re- 
bound. What  is  the  entire  distance  traveled  by  the  ball  until 
it  comes  to  rest  ? 

14.  Under  the  conditions  given  in  Ex.  13,  the  time  the  ball 
needs  for  the  first  ascension  and  fall  is  2  seconds,  and  the  time 
between  any  two  successive  rebounds  is  equal  to  the  preceding 
similar  period  multiplied  by  iV3.  In  what  time  does  the  ball 
come  to  rest  ? 


CHAPTER  XXII 

BINOMIAL   THEOREM 
PROOF  BY  MATHEMATICAL  INDUCTION 

368.   The  following  example  explains  the  demonstration  by 
mathematical  induction,  a  method  frequently  used  in  algebra. 
To  prove  that  ovV^^-HV 

4 

Assuming  that  the  proposition  was  correct  for  some  number 

Zj,  we  have  1-2 /x.,  1x2 

13^2^  +  3^. .■A;^  =  ^  V^  +  ^;  .  (1) 

Adding  (k  + 1)^  to  each  member, 

13  +  2»  +  33...(A;  +  lf  =  ^l(^±ll'+(A;  +  l)' 

4 

Or         l3  +  2^.|.33...(;^4.i)3^(fe  +  W  +  2f^  ^2) 

But  (2)  expresses  that  the  proposition  is  true  for  (A;4-l). 
Hence,  if  the  proposition  is  true  for  any  particular  number,  it 
must  be  true  for  the  next  higher  number. 

Evidently  the  proposition  is  true  for  k  =  l,  hence  it  is  true 
for  A;  =  2,  and  therefore  again  it  is  true  for  A;  =  3,  and  so  on  for 
higher  and  higher  numbers. 

As  this  mode  of  increasing  k  can  never  reach  an  end,  the 
proposition  is  generally  true. 

346 


BINOMIAL   THEOREM  347 

369.   The  method  of  proving  a  proposition  by  mathematical 
induction  consists  therefore  of  two  parts. 

(1)  Assu7ne  the  proposition  to  be  true  for  some  number,  and 
demonstrate  that  it  then  must  be  true  for  the  next  higher  one. 

(2)  Prove  that   the  proposition   is  true  for  some  particular 
number. 

EXERCISE   138 

Prove  the  following  identities  by  mathematical  induction : 

1.  1  +  2  +  3H [-n  =  ln(n  +  l). 

2.  2  +  4  +  6H [-2n  =  n(n-\-l). 

3.  12  +  22+32+. ..+n2  =  i7i(n  +  l)(2n  +  l). 

4.  1    +    1    +    1     +        .         1 


1.22.33.4  n(n  +  l)      n  +  1 


1.3     3.5     5.7  (2n-l)(27i  +  l)      2n  +  l 

12  3 

2.3.4'^3.4.5"^4.5.6"^"* 

, n_ n(n  +  l) 


(n  +  l)(ri  +  2)(n+3)      4(n  +  2)(n  +  3) 

7.  1.2  +  2.3  +  3.4+...+n(ri+l)  =  in(?i  +  l)(n  +  2). 

8.  1.2.3  +  2.3.4  +  3.4.5  +  ...  +  7i(n  +  l)(w  +  2) 

=  Jn(7i  +  l)(w  +  2)(n  +  3). 

9.  _i_  +  ^_  +  _A_+... 

3.4.54.5.65.6.7 


+ 


ri n(n-\-l) 


(n  +  2)(7i  +  3)(w  +  4)      6(n  +  3)(n  +  4) 

10.  l  +  2.2  +  3.22  +  4.23  +  ...+n.2''-^  =  (n-l)2«  +  l. 

11.  i  +  2.3  +  3.32  +  4.3^+.-  +  7i.3"-^  =  ^^^^~^^''^"+-. 

4 


3J:8  ELEMENTABY  ALGEBRA 

BINOMIAL  THEOREM  FOR  INTEGRAL  EXPONENTS 
370.    It  has  been  shown  in  §  220  that 

.  5  .  4  .  3  .  2  . 1  ^5 


1.2.3.4.5 


The  method  of  induction  furnishes  a  convenient  means  for 
proving  this  proposition  for  any  positive  integral  exponents, 
i.e.  to  prove 

^a^+n  .  a'^-ft  +  ^^zl)  a-^6^+  n{n-l)(n-2)  ^^_,^,  _^^^     ^^^ 

Assuming  that  the  proposition  is  true  for  a  number  fc,  we 
have 
(a  +  hf  =  a'+Jca^-'b  +  ^(f  ~^)  q.-2^2^^(fe -l)(fc-2)  ^,_s^,  ,,,^ 

Multiplying  both  members  by  (a  +  b),  and  collecting  similar 
terms  in  the  right  member, 


(a  +  by+''  ==  a^+1  +  (A;  4-  l)a'^>  + 


Z 


r  A:(A;-l)(fe-2)     A:(A;-1)  1    .2.3 ... 
■^1        1.2.3        "^     1-2     J 
Or  (a  +  by^^ 
==a^+i+(fe+l)a^6+  (^+^)^a^--&2^  (?r+l)^(A:-l)  ^.-2^3  ...^    (2) 

But  (2)  expresses  that  the  proposition  is  true  for  the  exponent 
(A;  +  1),  since  every  li  of  (1)  has  been  replaced  in  (2)  by  a  {k  +  1). 
Therefore,  if  the  proposition  is  true  for  any  exponent,  it  must 
be  true  for  the  next  higher  one,  and  since  it  is  true  for  A;  =  2, 
it  is  generally  true.     (§  369.) 


BINOMIAL   THEOREM  349 

371.  The  product  1  •  2  •  3  •••  n  is  called  factorial  n,  and  it  is 
frequently  represented  by  the  symbols  \n  or  n ! 

Thus  [3  =  1.2.3  =  6;  51  =  1.2.3.4.5  =  120. 

372.  Using   this    symbol,   the   binomial   theorem   may   be 

written, 

(a  +  hy  =  a"  +  naP-^h  +  ^^^~-^)  a^-^'h^  +  ... 

n{n-l)...{n-r  +  V)  ^„_,.^. ,,,  ^  ^^ 

The  student  should  note  that  every  coefficient  of  the  ex- 
pansion has  the  same  number  of  factors  in  numerator  and 
denominator. 

373.  The  general  term  of  expansion,  i.e.  the 

(r  +  l)th  tenn  =  n{n -I)  ••>  {n  -  n-^l)  ^„_,^, 

in 

Ex.1.     Expand  (2  a2_V6)^ 

(2a2-V6)5=(2a2_5')5 

=  (2  a2)5  _  5  .  (2  a2)452  ^  tii  (2  a^y^b^)^ 
\  •  2 

-  ^4^  (2  a2)2(5i)3  +  ^•4-3.2  ^2  «2)  (5^)4  _  (5^)6 
l«2.iJ  \  '  1  '  6  -  ^ 

=  32  aio  -  80  a^b^  +  80  a^ft  _  40  a^b^  +  10  a^b^  -  b^. 

Ex.  2.   Find  the  7th  term  in  the  expansion  of  (3  a  +  h^f. 

Since     r  +  1  =  7,  r  =  6,  and  n  =  8. 

Hence  the  7th  term  =  ^''i  -^'^'^'^  (3  a)2(52)6  =  262  a%^. 
1.2.3.4.5.6  ^      ^  ^    ^ 

Ex.  3.    Find  the  middle  term  of  (a;  -  2  y)^^. 

Since  the  expansion  has  11  terms,  the  6th  termjs  the  middle  term,  or 

r+l=6. 
Therefore  r  =  5,  w  =  10,  a  =  ic,  b  =  —  2y. 

Hence  the  middle  term  =  ^^'^'^•'^  -^  ^H-  2  yy. 
1.2.3.4.5      ^        ^^ 

=  -  8064  x^y\ 


2.    (x  +  Sy.  \b      a, 


350  ELEMENTARY  ALGEBRA 

EXERCISE   139 

Expand  the  following : 

1.  (x+yy.  ^    (^_^y,  7.  {x'-Zfy. 

^    Simplify  : 

9.    (Va  +  V6)^+(Va-V6y.    11.    (1  +  V^)' -  (1  -  V^)^ 
10.    (1+V/I/+(1-V<.  12.    (V2  +  x)«+(V2-ajy 

13.  Find  the  4th  term  of  (a  +  hf. 

14.  Find  the  9th  term  of  {a  -  hf^ 

15.  Find  the  3d  term  of  {l-{-xy. 

16.  Find  the  4th  term  of  (a  +  3  hy. 

17.  Find  the  5th  term  of  (a^  -  3  hy. 

18.  Find  the  7th  term  of  ( ^a  - 


a 

19.  Find  the  10th  term  of  (1  +  xy\ 

20.  Find  the  5th  term  of  (x  -  -^ 

\       x^ 

21.  Find  the  20th  term  of  (1  -  x-^'Y^, 

22.  Find  the  middle  term  of  (a  +  6)^ 

23.  Find  the  middle  term  of  {m  —  Tif. 

24.  Find  the  middle  term  of  [x 


x^ 

25.  Find  the  middle  term  of  {a^  —  ^  alPf. 

26.  Find  the  middle  term  of  (-^m  —  —J  • 

27.  Find  the  term  independent  of  ic  in  ix 

28.  Find  the  term  independent  of  a?  in  ioi? 


BINOMIAL   THEOREM  351 

374.  The  coefficients  of  the  expansion  of  (a +6)",  called  the 
binomial  coefficients,  are  represented  by  several  symbols,  e.g. 
"CV,  JX,  C;\  and  C). 

Using  the  first  symbol, 
5^  ^5-4 
'1.2* 

'         1.2 
«r  ^K^-1)  •••  (n  —  r-\-l) 
|r 

375.  The  binomial  theorem  may  then  be  written 

Making  a  =  1  and  h  =  x, 

(1  +  xY  =  1  +  "CiX  +  ^G^^  +  •  •  •  +  "CX-  •  •  +  x\ 

376.  The  (/•  +  1)  term  may  be  written 

377.  If  we  multiply  the  numerator  and  the  denominator  of 

"C,  by  \n-r, 

n(^  _n{n-V)  ...  (n-r  +  l)(n-r)  ...2.1 
\r  \n  —  r 


\L\i 


Substituting  n  —  r  for  r, 

W-r  — 


|yi  — r|r 
That  is,  '•0,  =  "a„_,. 

378.   Hence  the  binomial  coefficients  of  any  terms  equidistant 
from  the  end  and  the  beginning  are  equal. 

Thus,  »«(78  =  ^«(72  =  ^^5^  =  45. 

1  •  Ji 


352  ELEMENTARY  ALGEBRA 

379.  By  substituting  a  =  1  and  &  =  1  in  the  formula  for  the 
binomial  theorem,  we  have 

~  That  is,  the  sum  of  the  coefficients  of  the  expansion  of 
a  4-  6  is  2^ 

Similarly,  the  sum  of  the  coefficients  of  the  expansions  of 
(a  +  2  hy  =  3",  etc. 

380.  In  higher  algebra  it  is  proved  that  the  binomial  theorem 
is  true  for  negative  and  fractional  values  of  n,  provided  a  is 
greater  than  b.  The  expression,  however,  gives  an  infinite 
number  of  terms. 

Thus,    (l^xy  =  l  +  -lx  +  i-^^x'-\-^'~^^'~^  -a?"- 

381.  The  sum  of  two  successive  binomial  coefificients  is  a  binomial 
coefficient  of  the  next  higher  order. 

For    nn  +  nc,_,  =  ^^(^  -  1)  ...  (n  -  r  +  1)  ^  n(n  -  1)  .-  (n  -  r  +  2) 
1  •  2  •••  r  1  .  2  ...  (r  —  1) 

_n(n-l)  "•  (n-r  +  2)  (n-r  +  1      ^] 
1.2.3  ...  (r-1)        I         r  ) 

_n(n-l)  ...  (n-r  +  2)  ^  (n  +  1) 
1.2.3...  (r- 1)        *        r 

^(n  +  1)71  .  (n  -  1)  ...  (n-r  +  2)_^^,^ 
l-2'S".r  *"* 

E.g.  «C3  +  ^C2  =  6(73,   "Oe  + '^Os  =  "+1O6. 

382.  'The  proof  for  the  binomial  theorem,  as  given  in  §  370,  referred 
only  to  the  first  three  terms.  To  prove  that  it  is  correct  for  any  term  we 
may  use  the  proposition  of  the  preceding  paragraph.     Assuming  that 


BINOMIAL    THEOREM  353 

Multiplying  by  (1  +  ic), 

^  ^,      (-1+     "(7ia:+     "C2x2  ...  «Cr_i3C'-i+     ""CrX^     +...5c« 

(1  +  ic)"+i  =    1  +  "+iC'iX  +  «+i02ic''2  +  «+iC;a;'-     +  •••  x»+i. 

Hence  it  follows  by  the  method  used  in  §  369  that  the  theorem  is 
correct  for  all  terms. 

EXERCISE  140 

•    1.    Find  '^a^,  i^^Cioo. 

2.  Find  the  98th  term  of  (a  +  b)'^- 

3.  Find  the  47th  term  of  (1  -  x)"^. 

4.  Find  the  sum  of  the  coefficients  in  the  expansion  of 

5.  Find  the  sum  of  the  coefficients  in  the  expansion  of 

(l-xy. 

6.  Find  the  sum  of  the  coefficients  in  the  expansion  of 

7.  Find  the  sum  of  'C,,  '0„  'C„  'C^,  'C„  and  ^Ce  without 
finding  the  value  of  each  coefficient. 

8.  Findn,  if  "C2  =  28. 

9.  Find  w,  if  "Og-^  "02  =  10. 

10.  Expand  (1  +  x)~^  to  four  terms. 

11.  Expand  (1  —  x)-^  to  four  terms. 

12.  Expand  (1  —  x)^  to  four  terms. 

13.  Write  the  term  of  ( 1  +  —  J  ,  which  contains  x'^ 

14.  Write  the  term  of  [  aj  — ] ,  which  contains  ar^. 

15.  In  the   expansion  (a +  6)^",  the   coefficient  of   the  ?*th 
term  is  equal  to  the  coefficient  of  the  2rth  term.     Find  r. 

2a 


354  ELEMENTARY  ALGEBRA 

REVIEW  EXERCISE  V 
Simplify  : 

1.  (^_  6  2-2 +  27^-4)^  (J +  2^-1 +  3  2-2). 

2.  (7^r^^  +  A/^'. 

3.  (ii'x'-  -n{n-l) ar^"-i  +  (^  ~  ^)''a^H-2y. 

4.  2V|  +  V60-V15  +  V|. 

5.  7a/54 +3^16 +-\/2- 5^128. 

6.  (9  +  2VlO)(9-2VlO). 

7.  (2V8  +  3V5-7V2)(V72-5V20-2V2). 

8.  (V3-i-V2)-^(V3-V2). 

9.  1^(^2-1).  11.    (5-V24)^. 

10.  (7H-4V3)^.  12.    (a;  +  2Vx-l)^. 

13.  -^^ ^±2__^     1  1     . 


a?  —  x-^l      x^  -\-x-\-l      x  —  1      x  +  1 

a\x-h){x-c)      b\x-a)(x-c)     (?{x-a){x-h) 
'      (a-h)(a-c)   "^  (b-a){b-c)   "^   (G-a)(c-b)  ' 

a2  +  &2  _^  c2  +  2  a6  4-  2  ac  +  2  6c 
15.    


16. 


^2  _  52  _  ^2  _  2  5c 
Sa  —  5x-j-Say  —  5xy  f       2a  5 


24  a;  H- 16  aaj  +  12  a;?/ +  8  aa;?/  "    l2a.T  +  3a;     8a+12, 

17.  Find  the  coefficient  of  aW  in  (a  +  ?>)''. 

18.  Find  the  coefficient  of  a^  in  (a  -  l)^^'. 

19.  Find  the  middle  term  of  the  expansion  of  (a  —  by^ 

20.  Find  s    for  the  series, 


n  +  1      (n  +  iy      {n  +  ly 


BINOMIAL   THEOREM  356 

21.    Solve  a;  =  lH = — 

1+- 

X 


22.  Solve  ^^  +  9-^/^-9  =  3. 

23.  x^  =  l  —  x.     Find  x  to  four  decimal  places. 

24.  Solve  a^-^^-±^x-{- 1  =  0. 

ah 

25.  Solve  32 a^'^c""^  +  4 a"»+V-i {ac? -2)x  =  aV+^a^. 

26.  Find  the  equation  whose  roots  are  a  and  -• 

a 

27.  If  the  equation  x^ —  Sx-[-lb  =  k  has  equal  roots,  what 
is  the  value  of  A;  ? 


28.    Solve 


^  +  ^f  =  h  30.    Solve  |"  +  ^r"' 

4.x'  +  2f  =  2.  Vx^  +  f=h. 

ra^.y  +  a;/  =  30, 
29.    Solve  \  "     ""^^  "'  31.    Solve  n      1      5 


p-^  =  ^'  31.    Solve      11 

I  a;w  =  91.  -  +  -  = 

Ux  +  y)io?  +  f)  =  im, 
I  0^  +  2/^  = 


32.   Solve   ,    .       .     ^2g_ 


33.  A  body  projected  vertically  upward  with  a  velocity  of  v 

feet  per  second  reaches  after  t  seconds  an  elevation  s^vt  —  ^t"^. 
{g=  32  feet.)     If  v  =  64,  and  s  =  48,  find  t.  ^ 

34.  Find  three  numbers  such  that  the  product  of  any  two  of 
them  divided  by  the  third  one  gives  respectively  the  quotients 
a,  b,  and  c. 

35.  The  edges  of  two  hollow  cubes  differ  by  10  centimeters. 
If  a  certain  quantity  of  water  is  poured  into  the  larger  cube, 
there  remain  1578  cubic  centimeters  of  space  not  filled  with 
water.  If  the  second  cube  contained  142  cubic  centimeters 
more,  it  would  hold  the  quantity  of  water.  Find  the  number 
of  cubic  centimeters  of  water. 


356  ELEMENTARY  ALGEBRA 

36.  The  geometric  mean  of  two  numbers  is  to  their  arith- 
metic mean  as  3  :  5.     Find  the  ratio  of  the  two  numbers. 

37.  The  sum  of  five  numbers  in  A.  P.  is  2^,  and  their  prod- 
uct is  945.     Find  the  numbers. 

38.  A  man  spends  $6  on  the  first  day  of  January,  and  his 
expenses  increase  50^  every  day.  How  much  does  he  spend 
during  the  month  of  January  ? 

39.  The  population  of  a  town  increased  during  4  years  from 
10,000  to  14,641.  If  the  rate  of  increase  was  the  same  every 
year,  find  that  rate. 

40.  What  is  the  remainder  when  2a^H-3a^  —  2a;  is  divided 
by  a;  +  2  ? 

41.  For  what  value  of  I  is  2x^—^^+lx^—^x  +  l  exactly 
divisible  by  x  —  3  ? 

42.  Prove  that  a?"*  —  2  a;"  + 1  is  exactly  divisible  by  a;  —  1. 

43 .  Factor  6  a^  +  59  a;?/  + 105  y\ 

44.  Factor  (x^  _  5  ^j  -f  2)^  -  4. 

45.  Add  to  10  terms  x  +  y,  x^-^2y,  ar'  +  3^,  a;^  +  42/  •-. 

a;2  +  3  xy  =  7, 


46.  Solve    ,  09-.. 
xy  -\-6y^  =  14. 

47.  Solve  V^F^^==a}\ 

48.  Solve  (f )3--7  =  (jy^-3. 

49.  Factor  a2-a6  +  3a-2  62_354. 2. 

50.  Solve  the  equation  a;^  —  3  a;2«  + 1  =  0. 

51.  Simplify  22  +  100-4t-(^)-^  +  0^4-  \/S- 


52.    Solve 


a     0 
X     y 


BINOMIAL   THEOREM  357 

63.    Find  the  L.  C.  M.  of  x'  -x%  a^  -  x^,  a^  + 1. 
54.    Find  the  fifth  term  of 

\</b     3^ 
and  reduce  it  to  its  simplest  form. 


iW- 


55.    Simplify  -^  ^     ,-       3-     5/-.. 


CHAPTER   XXIII 
INEQUALITIES* 

383.  An  inequality  is  a  statement  that  one  quantity  is  greater 
or  less  than  another. 

384.  The  signs  of  inequality,  >  and  <,  are  respectively  read 
"  is  greater  than  "  and  "  is  less  than.'' 

The  members  or  sides  of  an  inequality  are  the  two  expressions 
which  are  connected  by  a  sign  of  inequality. 

Thus,  a>6  is  read  "a  is  greater  than  &."     a  is  the  left,  b  the  right, 
member. 

385.  One  number  a  is  greater  than  another  number  b,  if 
a  —  &  is  positive ;  similarly,  a  <  b,  it  a  —  b  is  negative. 

-5>-7,  since -5-(-7)  =  +  2. 

386.  The  symbols,  >,  <,  ^,  express  respectively  "is  not 
greater  than,"  "  is  not  smaller  than,"  and  "  is  not  equal  to." 

387.  Two  inequalities  are  of  the  same  species,  or  subsist  in 
the  same  sense,  if  their  signs  of  inequality  are  alike. 

a>  6,  c>  d,  are  of  the  same  species. 
4  >  3,  —  4  <  —  3,  are  of  opposite  species. 

388.  Tlie  sense  of  an  inequality  is  not  changed  if  both  members 
are  increased  or  diminished  by  the  same  number. 

Suppose  a>b,  then  a  —  6  is  positive. 
Hence  (a  +  m^  —  {b  +  m)  is  positive. 

Therefore  a  +  m^b  +  m. 

,    Similarly  a  —  m>b  —  m. 

*  This  chapter  is  not  required  for  the  examinations  of  the  College  En- 
trance Examination  Board. 

358 


INEQUALITIES  359 

389.  It  follows  from  the  preceding  paragraph  that  a  term 
may  he  transposed  from  one  member  of  an  inequality  to  another ^ 
by  changing  its  sign. 

390.  The  sense  of  an  inequality  is  not  changed  if  both  members 
are  midtiplied  or  divided  by  the  same  positive  number. 

Let  a  >  6,  and  m  be  a  positive  number. 

Then  a  —  6  is  positive. 

Hence  m{a  —  b)  or  ma  —  mb  must  be  positive ;  i.e.  ma  >  mb. 

In  a  similar  manner  it  can  be  proved  that  —  >  — 

mm 

,    391.   If  the  signs  of  all  terms  of  an  inequality  are  changed j 
the  sign  of  inequality  must  be  reversed. 

Consider  the  inequality  a  —  b-\-c'>x  —  y. 

Transposing  all  terms,       —x-{-y>  —  a-\-b  —  c. 

That  is,  —a-\-b  —  c<  —  x-[-y. 

392.  Tlie  sense  of  an  inequality  is  reversed  if  both  members 
are  multiplied  or  divided  by  the  same  negative  number. 

This  follows  from  §  390  and  §  391. 

393.  The  following  principles  can  easily  be  demonstrated : 

1.  If  any  number  of  inequalities  of  the  same  species  be  added, 
the  residting  inequality  will  be  of  the  same  species  as  the  original 
ones. 

2.  If  a>  b,  and  b>c,  then  a>c. 

3.  The  sense  of  inequalities  of  the  same  species  is  not  changed 
by  multiplying  their  corresponding  members,  p^'ovided  all  members 
are  positive. 

Hint.  If  a>&,  and  x>y^  then  ax>hx,  and  bx>by.  Therefore 
ax  >  by. 

4.  Tfie  sense  of  an  inequality  is  not  changed  by  raising  both 
numbers  to  the  same  power,  if  the  signs  of  the  members  are  not 
changed  by  the  involution. 


360  ADVANCED  ALGEBRA 

394.  An  identical  inequality  is  one  which  is  true  for  all  values 

of  the  letters  involved. 

Thus,  a  +  9  >  a,  and  m^  +  ti^  >  0,  are  identical  inequalities. 

395.  A  conditional  inequality  is  one  which  is  true  only  for 
certain  values  of  the  letters  involved. 

jc  +  1  >  5  is  a  conditional  inequality.     It  is  true  only  if  a;  >  4. 

396.  To  solve  a  conditional  inequality  for  a  certain  lietter 
means  to  find  all  values  of  the  letter  for  which  the  inequality 
is  true.  • 

Thus,  X  —  1  >  2  is  evidently  true  for  all  values  of  x  greater  than  3.  The 
value  3  is  sometimes  called  the  "limit  of  x "  in  the  inequality  x  —  1  >2. 

397.  Conditional  inequalities  are  solved  in  the  same  manner  as 
equations.  Care,  however,  should  be  taken  to  change  the  sense 
of  an  inequality  if  the  signs  of  both  members  are  changed  by 
multiplication,  division,  etc. 

Ex.  1.     Solve  the  inequality: 

6  —  5x     2a;  +  3j^      x  —  5 
12  6  3 

Clearing  of  fractions,  6-5x-4x-6>12-4x  +  20. 

Transposing,  — 5ic  —  4a;  +  4«>12  +  20. 
Uniting,  -  5  a;  >  32. 

Dividing  by  —  5,  x<—  6f. 

398.  Identical  inequalities  are  usually  proved  by  the  same 
method  as  identical  equations  (§  169).  The  fact  that  the 
square  of  any  real  number  is  positive  can  often  be  used  for 
such  proofs. 

Ex.  2.   If  a  and  b  are  unequal,  positive  numbers,  prove  that 

^  +  ->2- 
b     a 


INEQUALITIES  361 

The  inequality  is  true, 
if  a^  -\-  b^'>2ah  (clearing  of  fractions), 

or,  if  a^  —  2  a&  +  &2  >  0  (transposing), 

or,  if  (a  -  6)2  >  0, 

But  the  last  inequality  is  true. 

Hence  «4.^>2. 

6     a 

Note.     All  letters  in  the  present  chapter  are  supposed  to  represent 
real  numbers. 

399.   If  a  =  6,  then  a^+b^  =  2  ab ; 

if  a^-h,  \hQn  a'  +  W>2db. 

Hence  a?  +  W  is  either  equal  to  or  greater  than  2  a&,  a  state- 
ment that  may  be  written : 

a2  +  62^2a6,  (1) 

or,  o?  +  h^<f:2ah.  (2) 

Many  proofs  of  inequalities  can  be  based  upon  the  iden- 
tity (1). 

Ex.  3.   Prove  that  a^  +  5^  +  c^  ^  a6  4-  &c  +  ac. 

a2  +  &2  >  2  ah. 

52  4.  c2  ^  2  he. 

c2  +  a2  ^  2  ca. 
Adding,  2  a2  +  2  62  4.  2  c2  ^  2  a6  +  2  &c  +  2  ac. 

Dividing  by  2,  a2  +  62  +  c2  >  a6  +  6c  +  ac. 


362 


ADVANCED  ALGEBRA 


EXERCISE   141 
Solve  the  following  inequalities : 


1.   3x_5^2x^2. 
4  5 

^     x  —  2.x  —  4:^   x  —  3 


+ 


> 


„     x—S     x—8 ^5—x 

^'     ~~7i 7, —  <> Z — * 


4.  |  +  2<-13-|. 


'•    -li ^^-^• 

6.    x'+{x-\-l)(x-l)-2{x-2){x  +  3)>0, 

g      2(x-7)      x  +  2      x-\-l 
3  4     "^    11   * 

9.    (l  +  6^)2-(l4-10a:)2  +  (2+'8aj)2>0. 

10.  a?  +  -  >  5,  if  a  is  positive. 

a 

11.  (a  — a;)(&+^)>«^— a^,  if  a  — &>0. 

12.  -^^ <  -,  if  a  and  b  are  positive. 
1  —  a;      b' 

13.  Between  what  limits  must  x  lie,  if 
•:^I=i  +  ^±i<2,  and 

Xx  +  l)(x-{-2)>x'-^5? 

14.  Solve  a^_i69<0. 

15.  If  ^Lzl > 2a4i&^  ^^(j  J  -g  positive,  solve  for  a. 

b  8b 


16.    "Find  the  limit  of  x,  if 

( x-}-4:y  =  37,  and 
l2aj  +  52/>53. 


INEQUALITIES  363 

If  a,  b,  and  c  represent  unequal,  positive  numbers,  prove  that : 


17.    a'-\-3b^>2b(a  +  b).  ^^     a  +  b^     ah 

18     ^!!±^'>^  +  6^  ■  *       2         a  +  6 

4  2^'  20.    a%  +  ab^>2a%\ 

21 .  (a^  +  6^)  (a^  +  6^  >  (a«  +  bj. 

22.  a^  +  6^  >  a6  (a  +  6).      Hint.    Divide  by  a  +  6. 

23.  a  +  b>2^ab. 

24.  a?-b^>a%-ah\ 

25.  (a^  +  ?>')  (^'  +  c^)  (c^  4-  a^)  >  8  a^ftSc^. 

26.  (a  4- &)  (&  +  c)  (c  H- a)  >  8  a6c. 

27.  Wliich  is  the  greater,  o?b  +  a6"^  or  a''  +  6*  ? 

28.  Prove  that  d^  +  6»  +  c^  >  3  abc. 


CHAPTER   XXIV 
VARIABLES  AND  LIMITS 

400.  Functions  (§  292)  are  usually  denoted  by  symbols  of 
the  form  fix),  P{^)i  F(x),  etc.,  and  are  read  /  function  of  x, 
P  function  of  x,  etc. 

Thus,  f(x)  comprises  all  expressions  which  involve  the  letter  aj,  and  it 
may  represent  in  one  discussion  Sx'^  +  2aj  +  1,  in  another  Vx,  in  a  third 
4=»  +  2  x^,  etc. 

401 .  If  f(x)  is  known  in  any  particular  discussion,  f(a)  is 
formed  by  substituting  a  in  place  of  x. 

E.g.     If  /(ic)  =  2a;2  +  2a;  +  3, 

then  /(3)=  2.  32  + 2.3 +  3  =  27, 

/(a)  =  2  a2  +  2  a  +  3, 
/(O)  =  0  +  0  +  3  =  3,  etc. 

402.  If  y  =f(x),  X  is  called  the  independent  variable,  and  y 
the  dependent  variable. 


403.    Similarly,  fix,  y)  denotes  a  function  of  two  independent 

variables.     Thus,   a?  +  Zxy  +  y^j   or    V»  +  V?/,  or  -,  may  be 
represented  by  the  symbol  fix,  y). 

E.g.     If  /(x,?/)  =  a;y  +  x, /(2,3)  =  28  +  2  =  10. 


EXERCISE   142 

1.  If  f{x)  =  0^2  +  3  05  4-  2,  find  /(I),  /(O),  /(- 1). 

2.  lif{x)  =  bo?-2x- 3,  find  /(2),  /(a),  /(O). 

3.  If  fix)  =  4%  find  /(O),  /(-  1),  /a). 

364 


VARIABLES  AND  LIMITS  365 


4.  If  f(x)  =  -y/16,  find  /(2),  /(I),  /(4). 

5.  If  f(x)  =  {x-  m)ix  -  71),  find  /(m),  f(n),  /(O). 

6.  If /(x)  =  0^-2/2  and  F(x)  =  x-y,  find  ^^. 

7.  If  f(x,  y)  =  V^M^,  find  /(3,  4). 

8.  Solve  the  equation  f{x)  =f(J). 

9.  Solve  the  equation  f(x)  =f(a  + 1). 

10.  Simplify  -^^^^  +A-  ^)  +/(^)  -/(- '«). 

11.  If  /(a;)  =  a^  +  2  aj,  find  /(a  +  7i)  -f(a). 

12.  If  f{x)  =  0^  +  1,  find  /(aj  +  h)  -f(x). 

13.  If /(a;)  =  2a;  +  l,  find*fi^^±ii. 

14.  If  /(a;)  =  a%  find  /(^±-^. 

/(a;-/i) 


404.  The  limit  of  a  variable  is  a  constant  which  the  variable 
approaches  indefinitely  without  becoming  equal  to  it. 

The  difference  between  the  constant  |  and  the  variable  .6,  .66,  .666,  ••• 
is  respectively  less  than  ■^,  j^j^,  ystju^  '"-  ^Y  increasing  the  number  of 
decimals,  this  difference  can  be  made  smaller  than  any  assignable  number, 
but  the  decimal  can  never  equal  f . 

Hence  |  is  the  limit  of  .666  .-. 

405.  The  symbol  =  denotes  approaches  as  a  limit.  * 
Thus,  a;  =  a,  or  lim  x  =  a,  denotes  x  approaches  a  as  a  limit. 

406.  To  prove  that  a  variable  x  approaches  a  constant  a  as 
a  limit,  it  is  necessary  and  sufficient  to  prove  that  the  absolute 
value  of  a  —  a;  can  become  smaller  than  any  assigned  constant, 
however  small,  without  becoming  zero. 

E.g.    Let  X  equal  the  sum  of  n  terms  of  the  series 

1  +  i  +  i-... 


366  ADVANCED  ALGEBRA 

Then  a;^^-(i)-  =  2 L. 

1  -  i  2«-i 

Kence  2  —x  =  — — 

2n-i 

By  increasing  n  we  can  make  ~-^,  or  2  -  ic,  as  small  as  we  please. 

Thus,  to  make  2  -  ic  <  ^J^^,  make  w  =  11, 

tlien  2  -  X  =  —  =     ^    . 

2'»-i     1024 

Similarly,  to  make  2  -  x  < yooffooo^  let  n  =  21, 

then  2  —  x=  -^  = ,  etc. 

2«-i     1048576 

Hence  2  —x  can  be  made  smaller  than  any  assignable  number,  but 
2  —  X  cannot  be  made  equal  to  zero. 
Therefore  x  approaches  2  as  a  limit. 

407.  If  a  variable  increases  so  that  its  absolute  value  becomes 
greater  than  any  assignable  number,  however  great,  the  variable 
is  said  to  increase  indefinitely  or  to  become  infinite  (oo). 

Thus,  -  increases  indefinitely,  if  a;  =  0. 

X  •" 

408.  An  infinitesimal  is  a  variable  whose  limit  is  zero. 
Thus,  -  becomes  infinitesimal,  if  a;  =  c». 

X 

A  finite  number  is  a  number  which  is  neither  infinite  nor 
infiyiitesimal. 

409.  The  symbol  \_f(x)\^,  =  h,  or  lim  \_f{x)'],^,  =  h,  denotes 
f(x)  ap2)roaches  b  as  a  limit,  if  x  a2jproaclies  a  as  a  limit. 


00. 


Thus,  lim  f-l       = 


Or 

Note.     The  last  two  statements  are  frequently  written  in  the  abbrevi- 
ated form  y  . 

i  =  ooand— =0.     (§  209  and  §  210.) 

0  QO  ^  / 


VARIABLES  AND  LIMITS  367 

410.  If  tivo  variables,  x  and  /,  are  alivays  equal,  and  x 
approaches  a  as  a  limit,  then  y  approaches  a  as  a  limit. 

Since  a  —  x  —  a  —  y, 

(1)  a  —  y  can  be  made  smaller  than  any  assignable  number. 

(2)  a  —  y  cannot  equal  zero. 
Hence  y  =  a. 

411.  If  two  variables  are  always  equal,  and  each  approaches  a 
limit,  the  limits  are  equal. 

This  follows  directly  from  the  preceding  paragraph. 

412.  The  following  relations  are  frequently  used : 

(1)  lim  {x  +  ?/)  =  lim  x  +  Hm  y. 

(2)  lim  {xy)  =  lim  x  x  lim  y. 

(3)  lim  ( -  J  =  lij^  ^  -J-  lim  y. 

413.  Vanishing  fractions.  A  number  of  functions  assume 
indeterminate  values  for  certain  values  of  the  independent 
variable,  thus : 

£— L  =  2,ifa.  =  l. 

rz =  00  —  00,  ifa;  =  0,  etc. 

2x     X 

414.  The  principal  indeterminate  form  is  -.     All  others,  as 

^     0 

?^,  0^  00  —  00,  0°°,  00^  can  be  reduced  to  the  form  -. 

OO'        '  '         '  '  Q 

415.  Many  of  these  functions,  however,  approach  definite 
limits,  if  we  let  x  approach  its  assigned  value  as  a  limit. 
These  limiting  values  are  usually  considered  the  true  values  of 
the  functions. 


368  ADVANCED  ALGEBRA 

1  —X 

Ex.  1.   Find  the  limiting  value  of -,  if  ic  =  1. 

1  — ar 

Direct  substitution  produces  the  value  -,  but  by  reducing  the  fraction 

to  its  lowest  terms,  we  have, 


Li  -  x^jx^    Li  +  xL^   2 


+ 

Ex.  2.    Find  the  limiting  value  of 

-— -^ — ,    when  a?  =  00. 

2a;-3-3a;2  +  4' 

Direct  substitution  produces  ^ ;  hence  divide  numerator  and  denomina- 


tor by  x^t 


CO 


4-1  +  1 

4x^-2x  +  l  _        x^     x^ 

2a;3-3a;2  +  4~  2_§4.i* 
X     «3 


2      1 

If  jc  =  00,  — ,  — ,  etc.,  approach  0  as  a  limit. 

Hence  the  required  limit  is  -  or  2. 

Ex.3.    Find     limf^^ ^^^l    . 

Direct  substitution  produces  co  —  oo,  hence  simplify, 

x^  3a;-4_a;2-6y  +  8_a;-4_ 

2x-4       x-2~    2(ic-2)     ~     2 

If  X  =  2,  this  value  reduces  to  —  or  —  1. 

Ex.  4.    Find  r/(E±^ZLM1     ,  if  /(«,)  =  ^. 

(a;  -f.  /t)3  _  a;3  ^  3  x'^h  +  3  x/t2  +  ^s 
U  h 

=  3  x2  +  3  x/i  +  ^2. 

Hence  V(x+hy-x^^      ^3^2. 


VABIABLES  AND  LIMITS 


369 


EXERCISE  143 

Find  tlie  following  limiting  values : 
1. 


UJx-*' 


11. 


;c2-2aj- 


a^  + 


5x    i,. 


^2     r(2a.'-2)(3a^-3).(5a^-4y 
L      (x-2)(x-3){x-4) 

"■["^-^r-^'-'l,.''/w-^ 


2b 


CHAPTER  XXV 

IMAGINARY  AND  COMPLEX  NUMBERS 

ALGEBRAIC   TREATMENT  OF   COMPLEX  NUMBERS 

416.  Since  the  square  root  of  a  negative  number  cannot  be  a 
positive  or  a  negative  number,  it  becomes  necessary  either  to 
exclude  from  our  consideration  such  roots  as  V—  «,  V—  1,  or 
to  enlarge  the  boundaries  of  the  number  system  by  the  intro- 
duction of  a  new  kind  of  number,  called  imaginary  number 
(§226). 

417.  An  imaginary  number  is  an  indicated  square  root  of  a 
negative  real  number ;  as  V-^,  V— 5. 

We  assume  that  imaginary  numbers  obey  the  fundamental 
laws  of  algebraic  operation,  i.e.  the  commutative,  associative, 
and  distributive  laws  (§  51). 

418.  Every  imaginary  number  can  be  represented  as  the  prod- 
uct of  a  real  number  and  the  quantity  V—  1. 


For  V— a  =  Va(— 1)  =  V«' V— 1. 

419.  The  quantity  V—  1  is  called  the  imaginary  unit,  and  is 
usually  represented  by  the  letter  i. 

420.  The  form  a  V  — 1  or  ai  is  said  to  be  the  typical  form  of 
imaginary  numbers. 

Note.     In  the  entire  chapter  the  letters  a  and  b  represent  real  numbers. 

.421.    A  complex  number  is  the  sum  of  a  real  number  and  an 
imaginary  number. 

a  +  &i,  4  +  V—  3,  are  complex  numbers. 

370 


IMAGINARY  AND   COMPLEX  NUMBERS  371 

422.  Two  complex  numbers  are  said  to  be  conjugate  if  they 
(lifter  only  in  the  sign  of  their  imaginary  terms. 

a  +  bi  and  a  —  hi  are  conjugate  complex  numbers. 

Note.  The  term  imaginary  is  sometimes  used  in  a  wider  sense,  viz.  as 
including  imaginary  and  complex  numbers.  Numbers  of  the  form  aV— 1 
are  then  called  pure  imaginaries.  Using  the  term  in  a  wider  sense, 
imaginaries  may  be  defined  as  even  roots  of  negative  numbers. 

423.  Fundamental  principle.     From  the  definition  of  square 

root  (§  21),  we  have 

(V^-^-1,  or  /2  =  -l. 

Note.  Beginners  sometimes  expect  that  (V  — 1)^  should  equal  V(  — l)'-^ 
or  v'I=±l.  The  answer  +1,  however,  is  evidently  wrong  as  (  +  1)2;:^  — 1. 
The  error  has  been  produced  by  a  careless  application  of  the  law  (  Va)"* 
=  y/a^,  which  is  true  only  as  far  as  the  absolute  values  of  the  two  members 
are  concerned.  If  this  law  is  applied  without  restriction,  it  gives  wrong 
results  even  for  real  numbers  (§  272). 

E.g.    (V6)2  equals  6,  and  only  6  (§  21). 

But  the  law  applied  would  give 

(\/6)2  =  V36=±6, 
the  answer  —  6  being  evidently  wrong. 

424.  The  higher  powers  of  V—  1  are  found  by  repeated  multi- 
plication.   

i  =V-1. 

*2=-l. 

Since  i"*  =  1,  i^  must  equal  i,  i^  =  P,  etc.  If  r  denotes  an  in- 
tegral number, 

i»  =  i^-K 

E.g.    i^  =  i^-^  =  t^  =  --^^l'j  e^^  =  t^  =  -.l. 


372  ADVANCED  ALGEBRA 

425.    To  multiply  imaginary  numbers,  reduce  them  to  their 
typical  form. 

V—  a  X  V^  =  Va  V^^  X  Vb  V^3=  —  -Vab. 

Note.    If  we  had  applied  the  principle  Vm  x  Vn  =  Vnm  directly,  we 
should  have  obtained  a  wrong  answer,  viz.. 


V-a  X  V-&=V(-  a)(-  b)  =±V^. 

The  difficulty  is  here  caused  by  the  restriction  of  the  sign. 

If  -y/^  were  taken  in  its  true  meaning  ±  \A^,  then  ±  V^a  x  ±  V^ 
would  equal  ±  y/ah. 

But  for  practical  reasons  it  is  convenient  to  consider  V—  a  as  +  V—  a, 
and  then  V—  a  x  V—  h  =  —  Vab. 

It  should  be  borne  in  mind,  however,  that  this  difficulty  is  due  to  the 
arbitrary  restriction  of  signs,  and  not  to  the  exceptional  nature  of  the 
imaginaries. 


426.  General  principle.  Imaginary  numbers  (e.g.  V—  a^) 
should  be  reduced  to  their  typical  form  (i.e.  aV— 1)  before  they 
are  added,  subtracted,  multiplied,  divided,  etc. 

427.  The  results  of  examples  involving  imaginaries  should 
be  represented  in  the  typical  form,  a  +  bi,  i.e.  all  real  numbers 
should  be  combined  and  all  imaginaries  likewise. 

428.  Ifa-{-bi==c  +  di,  then  a  =  c,  and  b  =  d. 
For,  transposing,  we  have  a  —  c=(d  —  b)i. 

That  is,  a  real  number  is  equal  to  an  imaginary  number, 
which  is  impossible  unless  both  members  are  zero. 

Hence  a  —  c  =  0,  and  d  —  b  =  0. 

I.e.  a  =  c,  and  b  =  d. 


Ex.1.    Simplify  V^^49+V- 64- V- 7. 

VC:49+  V364-  V=T=7\/^T+8\/^^-V7\/^ 


IMAGINARY  AND   COMPLEX  NUMBERS  373 

Ex.  2.    Multiply  V^^  by  V^-28. 

v^^  X  v^^  =  V7 .  v^n: .  V28 .  v^n: = - 14. 


Ex.  3.   Add  4  +  V-16  and  o-V-25. 

(4  +  V'ric)  +  (5  -  V^^25)  =  4  +  4 1  +  5  -  6 i 
=  9-1. 

Ex.  4.    Multiply  4  + V^^  by  1+V^^. 

4  +  iVJ 
1  + W/ 

4  +  I V7 

-  7  +  4  t-  V7 

-3  +  5  i  V7,  or  -  3  +  5  V^^. 
Ex.  5.    Divide  2  +  5V^l  by'3-2V'^. 


2  +  5  i . 
3-2i 

_  2  +  5  i     3  +  2  i  _  -  4  4-  19 1  _      4       19 
3-2^  3  + 2i            13                13      13 

Ex.  6.    Extract  the  square  root  of  4  +  2V— 45. 

Applying  the  method  of  §  275, 

Let 

V4  +  2  V-  45  =  Va:  +  Vy. 

Squaring, 

4  +  2  V-  45  =  a;  +  2  Vxy  +  y. 

Hence 

x  +  y  =  4. 

2  Vxy  =  2  V-  45. 

Squaring  (2), 

x^  +  2xy  +  y^  =  16. 

Squaring  (3), 

^xy=-  180. 

(4) -(5), 

x2-2aj2/  +  2/2  =  i96. 

Extracting  square  root,                x  —  y  =  14. 

But 

x-\-y  =  i. 

Therefore 

OJ  =  9,  and  y  =  —  5, 

and 

V4  +  2  \/-45  z=  3  +  V-  5. 

(1) 

(2) 
(3) 
(4) 
(6) 


The  root  can  be  found  more  easily  by  applying  the  inspection  method 
(§  276). 


374 


ADVANCED  ALGEBRA 


EXERCISE   144 


Simplify : 


1.  V^^25,  V=T2l,  V^^=^,  V^=^,  V=T. 

2.  V-49i»2-|-V-64x^-a;V-121. 

3.  aV^-iV^=^25¥^  +  V-49al 

4.  V  —  4  a?-?/^'  4-  i»  V — 9  ?/-  —  2/ V  —  36  a;^. 

5.  V'^^+2V^^-V^=^. 

6.  a V— a?  — ft V— i»  + V— c'^ic. 

7.  (4  +  2V^4-(3+V^  +  (6-V"-^). 

8.  (3a  +  V-16&2)  +  (8a-V-25  62)  +  (V^^9F+2a). 

9.  V—  25a?  —  V—  492/  +  V— 4aj  —  8 V^. 

10.  (V-9a6-V-4a6)-(V-a6-V-4a6). 

11.  V^=^4-V^^"^4-V"^^^18-V^^25^. 

12.  (V^^y+(V^«  13.      Z  +  ?-2  4-^3  +  ^4. 

14.    ^2_^2i^-6i^  +  ^2« 

15.  (-v^^+(V^^-(-v^i:)«. 

16.    V^^-V^^^^l^.  18.    3V^^^-4V^^. 


^'-  V-^-V-3^- 


17.    V— 2a-V— 50a. 

20.    (3V^^  +  V^=38  +  V^=^-2V-72)  •  V^^. 


21.  Va— 6  •  V6  — a. 

22.  I V— n  •  V4n  4-7iV— ^  •  V— 18. 


23.   (3  +  V-25)(3-V-25).  24.    (3+ V-25)(4- V-36). 


IMAGINARY  AND   COMPLEX  NUMBERS 


375 


25.  (7-V-a)(G+V-(/). 

26.  (11-12  0(11-100. 


27.  (6-V-7a)(3-V-7a). 

28.  (V^=^  +  V=^)l 

29.  (1  +  V^l 

30.  (Va-2x-{-V—a-{-2xy. 

31.  VT+1  •  Vl  —  i. 


32.  V-63^V-9. 

33.  V^^-^V3. 

4 


34. 


35. 


38. 


39. 


40. 


41. 


42. 


43. 


3  +  2V-1 

S-2i 
3  +  2i' 

2  +  5^ 
Si 

i  +  V=^. 
l±i 

1-?:* 

V3  + 1 V2 


l  +  V-3 
11 


44. 


3_-V-2 

36.  64--(l+3V^^). 

37.  3--(V2  +  V^. 

47.    (V— m  +  n)^— (V— m  — n)^. 


V3-iV2 

n  +  iv 

45.  (1  +  0^ 

46.  (l+^y+ (1-1)3. 


48. 


49. 


1  +  1 


1  +  i   ■   1 


50.  1  +  V-i^l-V^ 


1-V-l     1  +  V-l 


V—  a  — V— 6 


51.    3±V^^3 


V^=:2 


2_V-1     2  +  V-I 


Extract  the  square  root  of; 


52.  4-2V-45. 

53.  3  +  4z. 


54.  -5  +  24i 

55.  2i. 


56.    I. 


376  ADVANCED  ALGEBRA 

GRAPHIC   REPRESENTATION   OF  COMPLEX  NUMBERS 

429.  The  term  imaginary  was  introduced  into  algebra  be- 
cause problems  whose  answers  involve  imaginaries  have  usually 
no  solutions,  and  it  was  inferred  therefrom  that  such  numbers 
could  not  exist.  This  view  of  imaginary  numbers,  however, 
is  erroneous.  Problems  have  often  no  solutions  when  the 
answers  are  real  numbers;  e.g.  a  problem  requiring  the  num- 
ber of  persons  in  a  room  can  have  no  solution  when  the 
answers  are  negative  or  fractional.  But  it  would  evidently 
be  absurd  to  infer  therefrom  that  negative  or  fractional  num- 
bers are  unreal  or  impossible.  Similarly,  a  problem  asking  for 
the  ratio  of  a  number  of  men  to  a  number  of  women  can  have 
no  solution  when  the  answer  is  an  irrational  number.  But  this 
does  not  prove  that  irrational  numbers  are  unreal. 

430.  To  illustrate  the  reality  of  imaginary  numbers,  the 
geometrical  method  which  was  used  for  the  representation  of 
positive  and  negative  numbers  (§  11)  may  be  employed. 

Let  XX'  be  a  fixed  straight  line,  and  0  a  fixed  point  in  the 

line. 

From  0  lay  off  a  series  of  equal  lengths  to  the  right  and  to 

the   left.       Then   any   line 

x'                 B'        O          B  Y 

terminating    in    0,  ^s  OB,    ^«— < , — ,_< 1 j— , — i , — ^ 

represents  a  number.     For        ~       "^ 

convenience  we  shall  sometimes  refer  to  the  extremity  of  the 

line  as  representing  the  number.     Thus,  "  point  B  "  represents 

the  same  number  as  line  OB. 

1.  Rational  numbers.  Rational  or  commensurable  numbers 
are  represented  by  certain  points  in  XX' ;  viz.  positive  integers 
by  the  points  of  division  on  the  right  of  0,  negative  integers 
by  the  points  of  division  on  the  left  of  0,  and  fractions  by  cer- 
tain points  between  the  points  of  division. 

2.  Irrational  numbers.  Not  every  point  in  XX',  however, 
represents  an  integer  or  a  fraction,  as  we  can  construct  certain 


IMAGINARY  AND   COMPLEX  NUMBERS  377 

lengths  which  cannot  be  expressed  by  integers  or  fractions. 
Such  points  represent  incommensurable  numbers.  E.g.  if  we 
lay  off  on  OX  a  line  OB  equal  to  the  hypotenuse  ^ 

of  a  right  triangle  whose  other  sides  are  equal  to     J    "/^ 
unity,  OB  represents  V2,  a  number  which  cannot  be      1/ 
equal  to  an  integer  or  a  fraction. 

For  assuming  that  —  =  V2,  where  m  and  n  have  no  common  factor, 
n 

we  would  have  —  =  2,  which  is  obviously  impossible,  as  m^  and  n^  have 
no  common  factor. 

Note.  While  it  is  impossible  to  find  integers  or  fractions  which  are 
exactly  equal  to  an  irrational  number,  we  can  find  fractions  which  differ 
from  the  given  surd  by  less  than  any  number  which  we  can  assign. 

Thus,  V2  differs  from  1.4,  1.41,  1.414  •••  respectively  by  less  than  .1, 
.01,  .001,  etc. 

Hence  we  may  consider  \/2  the  limit  of  the  fraction  1.41421  •••. 

Every  irrational  number  may  be  regarded  as  the  limit  of  a  variable 
rational  number. 

Thus,  every  real  number  is  represented  by  a  point  in  the  line 
XX',  and,  vice  versa,  every  point  in  XX'  represents  a  real 
number. 

3.  Imaginary  numbers  can  be  represented  by  points  without 
XX',  as  may  be  shown  by  the  following  consideration : 

If  Ai  be  taken  a  units  to  the  right  of  0  in  OX,  then  the  line 
OAi    represents    the    number    a, 
and  an  equal  line  OA^,  drawn  in 
the  opposite  direction,  represents 
—  a. 

Now  obviously  the  line  OAi 
can  be  brought  into  the  position 
OA3  by  a  rotation  through  an 
angle  of  180°  about  0  as  a  center, 

while  algebraically  +  a  is  transformed  into  —  a  by  multiplying 
by  —  1.     Hence  it  follows : 


Y 

y-^ 

M 

"X 

X'    A3 

0 

-r-A, 

"1 

378 


ADVANCED  ALGEBRA 


431.  The  multiplication  of  a  real  number  by  —  1,  is  represented 
graphically  by  a  rotation  through  an  angle  of  180°. 

It  is  customary  to  rotate  lines  counter-clockwise,  i.e.  in  a  direction 
opposite  to  the  motion  of  the  hands  of  a  clock. 

432.  To  determine  the  algebraic  meaning  of  a  rotation 
through  an  angle  less  than  180°,  let  a  rotation  through  an  angle 
of  90°  represent  the  multiplication  by  an  unknown  number  x. 


Then 

OA2  =a  •  X, 

-Y 

and 

0As=0A2-x  = 

a 

X 

X. 

A2 

Or 

—  a  =  a-a^. 

/ 

/^'" 

N 

Therefore 

x  =  V-l, 
OA,  =  aV-l. 

( 

and 

X'   A3 

b~ 

-T— ^1   'X 

433.  We  may  therefore  consider  a  multiplication  by  V— 1 
to  be  represented  by  a  revolution  through  a  right  angle  counter- 
clockwise. 


434.  The  numbers  i,  2  i,  3 1,  etc.,  may  be  represented  respectively 
by  the  distances  1^  2,  3,  etc.,  laid  0^  on  OYj  which  is  perpendicu- 
lar to  XX'. 

lY 


X'-3-2-1 


31 

2i 

i 

0  , 


21 
3i 

1y' 


Similarly,  —  i,  —  2  i,  —  3  i,  are  represented  on  the  line  0  F'. 


IMAGINARY  AND   COMPLEX  NUMBERS  379 

The  four  quantities,  3,  3  z,  —  3,  —  3  i,  have  the  same  absolute  vahie, 
viz.  3,  and  each  is  represented  by  a  line  consisting  of  three  units,  but 
extending  in  different  directions,  viz,  : 

+  3  indicating  3  units  to  the  right, 

-1-  3  i  indicating  three  units  up, 

—  3  indicating  3  units  to  the  left, 

—  Si  indicating  3  units  down. 

435.  The  line  XX'  is  called  the  axis  of  real  numbers;  YY'  is 
called  the  axis  of  imaginaries.     The  point  0  is  called  the  origin. 

436.  Graphical  addition.  Two  real  numbers,  OA  and  OB, 
are  added  graphically  by  drawing  from  A  a  line  AC  equal  to 


-^ — I 1 ^ 


and  extending  in  the  same  direction  as  OB.     The  line  OC  is 
the  required  sum. 

E.g.    To  add  +  5  and  —  3,  lay  off  OA  =  5  to  the  right,  and  from  A, 
lay  off  ^O  =  3  to  the  left.     00  or  2  is  the  required  sum. 

437.    Imaginaries  and  imaginaries,  or  real  numbers  and  imagi- 
naries, are  added  graphically  in  the  same 
manner  as  real  numbers.     E.g.  to  add 

4  and  +  3  i,  lay  off  0A  =  -\-  4.  From 
A  draw  AC—^  upward,  i.e.  equal  and 
parallel  to  line  3  i,  or  OB.  OC  repre- 
sents the  required  sum. 

Note.     It  should  be  noted  that  4  +  3  i  is 
represented  by  the  length  and  direction  of  OC.     The  length  of  OC  is  5, 
but  it  would  be  erroneous  to  assume  that  4  +  3  t  equals  5.    The  numbei 

5  is  the  absolute  value  of  4  +  3  i,  but  it  is  not  equal  to  4  +  3  i. 


380 


ADVANCED  ALGEBRA 


438.  Complex  numbers.  If  a  and 
b  are  real  numbers,  the  complex 
number  a  +  bl  may  be  represented 
by  OB,  the  sum  of  a  and  bi.  I.e. 
Draw  OA  =  a,  and  AB  equal  and 
parallel  to  bi.     OB  represents  a-\-bi. 

E.g.  OE  represents  —  4  —  2  i. 


439.  The  absolute  value  or  modulus  of  any  number  (i.e.  real, 
pure  imaginary,  and  complex)  is  the  length  of  the  line  which 
represents  the  number.     It  is  always  taken  as  positive. 


The  absolute  value  of  a  +  bi  =  OB,  or  +  Va^  +  b^. 
The  absolute  value  of  -  4  -  2  z  =  \/42  +  2^  =  2  V5. 


440.   The  amplitude  of  OB  is  the  angle  XOB,  i.e.  the  angle 
between  OX  and  OB,  measured  from  OX  counter-clockwise. 

Ex.  1.   Determine  the  algebraic  meaning  of  the  rotation  of  a 
line  through  an  angle  of  60°. 


Let 
and 


OA=OB=OC=OD  =  lj 
ZAOB  =  Z  BOG  =  Z  GOD  =  60°. 


If  X  is  the  number  which,  applied  as  a  factor,  produces  the  required 
rotation, 
then  OB  =  x,   OG  =  x^,   OD  =  x\ 

I.e.  x^  =  -\, 

or  x  =  V^. 

The  rotation  through  an  angle  of  60°  represents  therefore  a  multiplica- 
tion by  v/—  1,  and  line  OB  represents  v^  — 1.  A  simple  geometrical  deduc- 
tion shows  that  OB  or  V^^  =  |  +  |V3  •  i. 


IMAGINARY  AND   COMPLEX  NUMBERS 


381 


Ex.  2.   Construct  graphically  the  different  values  of  -y/i. 

From  0,  draw  five  lines,  OA,  OB,  OC, 
OD,  and  OE,  each  equal  to  1,  and  forming 
angles  of  72°  with  the  two  adjacent  lines. 
If  OA  lies  on  the  axis  of  real  numbers, 
OA,  OB,  00,  OD,  and  OE  represent  the 
5  values  of  Vi.  (The  proof  is  similar,  to 
Ex.  1.)  By  actual  measurement  we  find 
5  values  1,  .31  +  .95  i,  -  .95  +  .31  i, 
-.95 -.31 1,   .31-.95i. 


EXERCISE  145 
Eepresent  the  following  numbers  graphically : 

1.  2^-4^.  3.-2  +  21  5.    _2-31 

2.  4-3?.  4.    i-1.  6.    -2i  +  5. 

7.  Construct  5  +  4  i  and  multiply  graphically  by  V—  1. 

8.  Prove  that  the  following  numbers  have  equal  absolute 
values :  4  +  3  I,  —  3  —  4  i,  —  5,  and  —  4  -|-  3  i. 

9.  Which  has  the  greater  absolute  value,  —  8  or  5  —  6 1  ? 

10.  Add  graphically  4,  3  i,  and  —  2. 

11.  Add  graphically  —  5,  —  2  /,  +2,  +5  ^. 

12.  Solve  the  equation  x^ -i-2x-\-  2  =  0,  and  represent  the 
roots  graphically. 

13.  Solve  the  equation  a^  — 1=0,  and  represent  the  roots 
graphically. 

14.  Solve  the  equation  flj^  + 1  =  0,  and  represent  the  roots 
graphically. 

15 .  Divide  2  i  by  1  +  i,  and  represent  the  quotient  graphically. 

16.  What  rotation  is  equivalent  to  a  multiplication  by  —  ^? 

17.  Find  graphically  V7. 

18.  Find  one  value  of  V— 1. 


382 


ADVANCED  ALGEBRA 


19.  Construct  7  lines  repref?enting  tlie  values  of  Vl. 

20.  Represent  graphically  iV2  +  -V2. 

21.  Eepresent  graphically  V3  +  i. 


441.    Complex  numbers  are  added  graphically  in  the  same  man- 
ner as  real  and  imaginary  numbers. 

E.g.  to  add  the  numbers  represented  by  OA  and  OB,  draw 
AC  equal  and  parallel  to  OB ;  then  00  represents  the  required 
sum.  (00  may  also  be  constructed 
by  drawing  the  diagonal  of  the 
parallelogram  determined  by  OA 
and  OB.) 

If  OA  =  a-\-  bi,  and  OB  =  c -{-  di, 
then  their  sum  is  a  -^  c  -\-  (J)  -\-  d)i  \  and 
in  order  to  demonstrate  that  OC  repre- 
sents the  required  sum,  we  have  to  show 
that  the  coordinates  of  C,  i.e..,  OE  and 

EC,  are  respectively  equal  to  a  +  c  and  h  +  d.  If  AD  is  drawn  parallel 
to  OX,  it  follows  easily  that  AOBF=  AACD.  Therefore  AD  =  c, 
CD  =  dy  and  since  OH  =  a  and  DE  =  AH  =  b, 


Hence 


O^  =  a  +  c  and  EC  =  b-\-do 
OC  =  a  +  c-}-{b  +  d)i. 


Note.  The  student  should  bear  in  mind  that  the  geometric  addition  of 
lines  which  have  length  and  direction  is  fundamentally  different  from  the 
usual  addition  of  lines  which  have  length  only.  The  length  of  00  is 
obviously  less  than  the  sum  of  the  lengths  of  OA  and  OB,  but  the  complex 
number  represented  by  OC  is  equal  to  the  sum  of  the  complex  numbers 
represented  by  OA  and  OB.  A  line  which  has  a  given  length  and  direc- 
tion is  called  a  vector. 

442.    Graphical  subtraction  of  complex  numbers. 

If  the  number  represented  by  OC  (diagram  of  §  441)  is  the 
sum  of  the  numbers  represented  by  OA  and  OB,  then  OA  is  the 


IMAGINARY  AND   COMPLEX  NUMBERS 


383 


difference  (tf  the  numbers  represented  by  OC  and  OB.  Hence 
to  subtract  OB  from  00  (see  an- 
nexed diagram)  draw  CA  equal 
and  parallel  to  OB,  but  extend- 
ing in  opposite  direction;  then 
OA  represents  the  required  differ- 
ence. 


Ex.  1.    Solve  the  equation 

and  construct  graphically  the  difference  of 
the  roots. 


By  formula, 


Draw  OA  =  1  -l- 1,  OB  =  1  —  i,  and  make  AC 
equal  and  parallel  to  OB,  but  extending  in  opposite 
direction.     0C  =  2i  is  the  required  difference. 


2i 
i 

\ 

0 

1    >  ^ 

1        2    *'' 

-i 

'^\ 

-2i 

Ex.  2.  Solve  the  equation  «^  — 64  =  0,  and  add  the  6  roots 
graphically. 

Factoring,  (x-2){d' +  2x  +  ^)(x  +  2)(af-2x-\-4:)  =  0. 

Hence  a?  —  2  =  0, 

aP^2x-\-4:  =  0, 
a;  +  2  =  0, 

ic2_2aj  +  4  =  0. 

These  equations  produce  the  following  roots  :  2,  —  1  ±  V  — 3, 
^  2,  1  ±  V^^,  or_  arranged  2,  1  +  iV3,  -  1  +  i V3,  -  2, 
—  1  —  WS,  1  —  iV3.  Assuming  V3  =  1.73,  Ave  obtain  respec- 
tively the  6  lines  OA^,  OAo,  OAq,  OA^,  OA5,  and  OAq. 


384 


ADVANCED  ALGEBRA 


To  add  these  quantities, 
make 

A1B2  li  and  =  OA2, 
B^B^  II  and  =  OA^, 
B^B^  II  and  =  OA^ 
B^Bq  II  and  =  OA^, 
BsBq  II  and  =  OA^. 

Since  Bq  coincides  with  0, 
OBq,  or  the  required  sum, 
equals  zero. 

(The  6  roots  of  the  equation  could  be  obtained  more  easily 
by  considering  that  x  =  ^64,  or  x=2^1.  Hence  construct  the 
6  values  of  \/i,  and  double  these  lines.) 


EXERCISE   146 

1.  Construct  4  +  3  ^  and  5  —  6 1,  and  add  them  graphically. 

2.  Draw  three  lines  representing    complex    numbers,   and 
add  them  graphically. 

3.  Divide  8  —  4 1  by  S  -\-i,  and  add  quotient  and  dividend 
graphically. 

4.  Divide  l-\-7  i  by  S-^i,  and   add   divisor   and  quotient 
graphically. 

5.  Multiply  2  —  3  i   by  3  +  1,  and  subtract  graphically  the 
first  factor  from  the  product. 

6.  Solve  the   equation   x'-^x-\-l  =  0,   and   add  the   roots 
graphically. 

7.  Solve  the  equation  a^  —  8  =  0,  and  add  the  roots  graphi- 
cally. 


IMAGINARY  AND   COMPLEX  NUMBERS  385 

8.  Show  graphically  that  the  sum  of  two  conjugate  complex 
numbers  is  a  real  number. 

9.  If  OA  represents  a  given  complex  number  a  +  hi,  con- 
struct the  number  2{a  +  hi). 

10.  Graphically  subtract  1  from  i,  and  add  —  l  +  i  to  the 
difference. 

11.  Draw  three  lines,  OA,  OB,  and  OC,  representing  any 
complex  numbers,  and  from  the  sum  of  the  first  two  subtract 
the  sum  of  the  last  two. 

12.  Draw  any  five  lines  representing  complex  numbers,  OA^, 
OA2,  OA^,  OAi,  and  OA5,  and  add  them  graphically. 

13.  If  OA  and  OB  represent  two  complex  numbers,  graphi- 
cally multiply  OA  by  i,  and  subtract  the  product  from  OB, 


2o 


CHAPTER   XXVI 
PERMUTATIONS   AND  COMBINATIONS 

443.  Fundamental  principle.  If  one  thing  can  he  done  in  m 
differeiit  ivays,  and  after  it  has  been  done,  a  second  thing  can  be 
done  in  n  different  ways,  then  the  two  things  can  be  done  together 
in  m  X  n  different  ivays. 

For  after  the  first  thing  is  done  in  any  one  way,  it  can  be 
associated  with  each  one  of  the  n  different  ways  of  doing  the 
second  thing,  and  thus  each  way  of  doing  the  first  thing  pro- 
duces n  ways  of  doing  the  two  things  together.  But  there  are 
m  ways  of  doing  the  first  thing,  hence  the  total  number  of  ways 
of  doing  the  two  things  together  is  m  X  n. 

Ex.  1.  An  alphabet  contains  5  vowels  and  21  consonants. 
How  many  different  words  of  two  letters  can  be  formed  from  it 
so  that  the  first  letter  is  a  consonant  and  the  second  a  vowel  ? 

The  first  letter  may  be  selected  in  21  different  ways.  Every  one  of 
these  letters  can  be  put  together  "tsrith  5  vowels,  or  each  consonant  produces 
5  different  words.     Hence  21  consonants  produce  21  x  5  or  105  words. 

Ex.  2.  Suppose  there  are  8  railroad  lines  operating  between 
Chicago  and  New  York,  and  9  steamship  lines  between  New 
York  and  Liverpool.  In  how  many  different  ways  can  a  man 
travel  from  Chicago  to  Liverpool  by  way  of  New  York  ? 

He  can  make  the  trip  from  Chicago  to  New  York  in  8  ways,  and  each 
of  these  may  be  combined  with  the  9  different  ways  of  going  to  Liverpool. 
Hence  he  can  make  the  whole  journey  in  8  x  9  or  72  different  ways. 

444.  By  successive  application  of  §  443,  it  can  easily  be 
shown  that  the  principle  is  true  if  there  are  three  or  more 
things  each  of  which  can  be  done  in  a  given  number  of  ways. 

386 


PERMUTATIONS  AND   COMBINATIONS  387 

Ex.  3.  In  how  many  different  ways  can  3  persons  be  seated 
in  a  coach  which  has  4  seats  ? 

The  first  person  can  be  seated  in  4  different  ways.  After  the  first  per- 
son is  seated,  the  second  can  be  seated  in  3  different  ways,  and  finally  the 
last  person  can  be  seated  in  2  different  ways.  Hence  the  total  number  of 
ways  is  4  X  3  X  2  =  24. 

EXERCISE  147 

1.  A  building  has  6  entrances.  In  how  many  different  ways 
can  a  person  enter  the  building  and  leave  by  a  different  door  ? 

2.  Between  two  cities  6  ferryboats  are  plying.  In  how 
many  different  ways  can  a  man  travel  from  one  city  to  the 
other  and  return  by  a  different  boat  ? 

3.  A  man  has  5  pairs  of  trousers,  7  vests,  and  6  coats.  In 
how  many  different  costumes  can  he  appear  ? 

4.  In  how  many  different  ways  may  an  English,  a  French, 
and  a  German  book  be  selected  from  6  English,  5  French,  and 
3  German  books  ? 

5.  In  how  many  different  ways  can  2  persons  be  seated  in  a 
coach  that  has  6  seats  ? 

6.  In  how  many  different  ways  can  3  children  be  seated  in 
3  seats  ? 

7.  How  many  different  words  of  two  letters  can  be  formed 
with  the  letters  a,  6,  c,  d,  and  e  if  the  first  letter  is  to  be  a 
vowel  ? 

PERMUTATIONS 

445.  The  permutations  of  a  certain  number  of  things  are  the 
different  orders  in  which  some  or  all  of  the  things  can  be 
arranged. 

Thus,  the  permutations  of  the  letters  a,  b,  c,  taken  two  at  a 
time,  are  ab,  ac,  ba,  be,  ca,  cb,  and  their  permutations,  taken 
three  at  a  time,  are  abc,  acb^  bac,  bca^  cab,  cba. 


388  ADVANCED  ALGEBRA 

Ex.  1.   Write  all  permutations  that  can  be  formed  from  the 
numbers  1,  2,  3,  4,  all  taken  at  a  time. 


1234 

2134 

3124 

4123 

1243 

2143 

3142 

4132 

1324 

2314 

3214 

4213 

1342 

2341 

3241 

4231 

1423 

2413 

3412 

4312 

1432 

2431 

3421 

4321 

Explanation.  "Write  first  all  permutations  whose  first  place  is  1,  then 
all  whose  first  place  is  2,  etc.  The  four  columns  represent  the  four  groups 
thus  obtained.  Each  group,  again,  is  divided  according  to  the  number  in 
the  second  place ;  thus  the  first  column  contains  first  the  permutations 
commencing  with  12,  then  those  commencing  with  13,  and  last  with  14. 
By  continuing  this  mode  of  arranging  the  permutations,  it  is  easy  to 
obtain  them  all. 

Ex.  2.  Write  all  permutations  of  letters  a,  6,  c,  and  d,  three 
taken  at  a  time. 


abc 

bac 

cab 

dab 

ahd 

bad 

cad 

dae 

acb 

bca 

cba 

dba 

acd 

bed 

cbd 

dbc 

adb 

bda 

cda 

dca 

adc 

bdc 

cdb 

deb 

446.  The  number  of  permutations  of  n  different  things 
taken  r  at  a  time  is  denoted  by  the  symbol  "P^.  Thus  the 
number  of  permutations  of  the  preceding  example  is  ^Fg. 

This  number  could  be  obtained  without  writing  all  permuta- 
tions. There  are  three  places  to  be  filled  in  by  letters.  The 
first  place  can  be  filled  by  a,  b,  c,  or  d,  i.e.  in  4  different  ways 
(represented  in  4  different  columns).  After  the  first  place  is 
filled  the  second  place  can  be  filled  by  one  of  the  remaining 
letters,  i.e.  in  3  different  ways  (producing  the  3  parts  of  each 


PERMUTATIONS  AND  COMBINATIONS  389 

column).     Similarly,  the  third  place  can  be  filled  in  2  different 
ways.     Hence,  according  to  the  fundamental  principle, 

447,  To  find  the  number  of  permutations  of  n  different  elements 
taken  r  at  a  time.  There  are  r  places  to  be  filled.  The  first 
place  can  be  filled  in  any  of  the  n  ways,  and  after  this  has 
been  filled,  the  second  place  can  be  filled  m  (n  —  1)  ways. 
Hence,  according  to  the  fundamental  principle,  the  two  places 
together  can  be  filled  in  n  (?i  —  1)  different  ways  j  or 

^P2  =  n(n-1). 

After  the  first  two  places  are  filled,  the  third  one  can  evi- 
dently be  filled  in  (n  —  2)  different  ways ;  or 

"P3  =  7i(7i-l)(n-2). 

By  continuing  this  process,  it  can  be  seen  that  "P^  is  equal 
to  a  product  of  r  factors,  the  first  factor  being  n,  and  each  fol- 
lowing factor  being  less  by  one  than  the  preceding  one.  Since 
the  last  factor  must  be  (n  —  r  + 1),  we  have 

"P,  =  n{n-l)(n-2)  ...  (/i-r  +  l). 

448.  The  number  of  permutations  of  n  elements,  taken  all 
at  a  time,  is 

n{n  —  l)(n  —  2)  •»•  to  w  factors. 


Or 


^P„  =  n(n-l)(n-2)    •'2-l=:\n 


Note.     Unless  stated  otherwise,  the  things  are  supposed  to  be  different 
and  not  to  be  repeated  in  one  permutation. 

Ex.  1.   How  many  different  permutations  can  be  made  by 
taking  4  of  the  letters  of  the  word  fraction  f 

8p^  =  8.7.6.5  =  1680. 


390  ADVANCED  ALGEBBA 

Ex.  2.  How  many  numbers  between  1000  and  10,000  can  be 
formed  with,  the  figures  1,  2,  3,  4,  5,  6,,  7,  no  figure  being  re- 
peated ? 

Since  numbers  between  1000  and  10,000  have  four  places,  the  required 
number  is 

7P4  =  7  . 6  .  5  . 4  =  840. 


EXERCISE   148 

1.  Write  all  permutations  of  the  numbers  1,  2,  3,  taken  all 
at  a  time. 

2.  Write  all  permutations  of  the  letters  a,  b,  c,  and  d,  taken 
all  at  a  time. 

3.  Write  all  permutations  of  the  letters  a,  b,  c,  d,  taken  two 
at  a  time. 

4.  Write  all  permutations  of  the  numbers  1, 2, 3, 4, 5,  taken 
two  at  a  time. 

5.  Find  the  value  of  'P^,  «Pi,  ^^Pg. 

6.  Find  the  value  of  ^Pg,  ^P^,  ^P^. 

7.  In  how  many  different  ways  can  5  pupils  be  seated  in  5 
seats  ? 

8.  In  how  many  different  ways  can  5  pupils  be  seated  in 
6  seats  ? 

9.  How  many  different  words   can   be   formed   with   the 
letters  of  the  word  equation  ? 

10.  How  many  different  numbers  of  three  different  figures 
can  be  formed  from  the  digits  2,  3,  4  ? 

11.  How  many  different  numbers  of  four  different  figures 
can  be  formed  from  the  digits  1,  2,  3,  4^  5,  6  ? 

12.  How  many  different  words  of  three  letters  can  be  formed 
from  the  letters  a,  b^  c,  e? 


PERMUTATIONS  AND  COMBINATIONS  391 

13.  In  how  many  different  ways  can  6  persons  be  placed 
in  G  seats  ? 

14.  Six  persons  enter  a  car  in  which  there  are  10  seats.    In 
how  many  different  ways  may  they  take  their  places  ? 

15.  How   many   different   arrangements   can   be   made   by 
taking  3  letters  of  the  word  theory  f 

16.  How  many  three-lettered  words  can  be  made  from  10 
letters,  no  letter  being  repeated  in  the  same  word  ? 

17.  In  how  many  different  ways  can  5  persons  form  a  ring  ? 

18.  If  "P3  =  4"P2,  findn. 


449.  To  find  the  number  of  permutations  of  things  which  are 
not  all  different,  taking  them  all  at  a  time,  let  us  suppose  the 
number  of  permutations  which  can  be  formed  by  taking  all  the 
letters  a,  a,  a,  h,  c,  was  x.  If  we  should  replace  the  three  equal 
a's  by  three  different  letters,  as  ai,  otg)  %>  the  number  of  permu- 
tations would  obviously  be  greater  than  x,  for  in  each  of  the  x 
permutations  we  could  arrange  the  a,  a,  a  in  [3  different  ways 
without  changing  the  position  of  the  b  and  c. 

Thus,  abaac  would  produce  the  6  permutations, 

a^a^asP,  a<}>a]a^G,  a^a^aoC,  a^ha^a^,  a^ba^a-fi,  ajba^a-fi. 

Similarly,  every  one  of  the  x  permutations  would  produce 
[3  arrangements,  i.e.  the  total  number  of  arrangements  would 
be  aj .  [3. 

But,  on  the  other  hand,  this  number  represents  the  number 
of  permutations  of  5  elements,  all  being  different,  or  [5. 


Hence  a?  •  [3  =  [5. 

13' 


|5 
Therefore  x  = .— • 


392  ADVANCED  ALGEBRA 

450.  In  general,  the  number  of  permutations  of  n  things,  of 
which  p  are  alike,  q  others  are  alike,  and  r  others  are  alike, 
taking  them  all  at  a  time,  is 

la 

Ex.  1.  In  how  many  different  ways  can  the  letters  of  the 
word  degree  be  arranged  ? 

The  word  contains  six  letters,  three  of  which  are  alike ;  hence  the 
number  of  permutations  is 

P  =  !=  =  120. 

Ex.  2.  In  how  many  different  ways  can  the  letters  of  the 
word  combination  be  arranged  ? 

Ill 
^-      •—      =4989600. 


L2^[2 


451.  Examples,  which  are  not  special  cases  of  the  general 
forinulve,  for  permutations,  should  be  solved  by  means  of  the 
fundamental  principle  (§  443). 

Ex.  1.  In  how  many  different  ways  can  the  letters  a,  b,  c,  d,  e 
be  arranged  so  that  the  first  letter  and  the  last  shall  be  always 
consonants  ? 

The  first  place  can  be  filled  in  3  different  ways,  and  after  this  is  done, 
the  last  place  can  be  filled  in  2  different  ways.  The  remaining  letters 
can  in  every  case  be  arranged  in  [3  different  ways. 

Hence  the  required  number  is  3  •  2  .  [3  =  37. 

Ex.  2.  A  boat's  crew  consists  of  8  men,  of  whom  3  can  row 
only  on  the  port  side,  and  2  only  on  the  starboard  side.  Find 
the  number  of  ways  in  which  the  crew  can  be  arranged. 

The  first  of  the  men  who  can  row  only  on  the  port  side  can  be  placed 
in  4  ways,  the  second  one  in  3  ways,  the  third  one  in  2  ways.  Similarly, 
the  two  men  who  can  row  only  on  the  starboard  side  can  be  placed 


PEBMUTATIONS  AND  COMBINATIONS  393 

respectively  in  4  and  3  different  ways.      The  remaining  men  can  be 
placed  in  [3  ways.     Hence  the  total  number  of  arrangements  is 

4  .  3  .  2  .  4  .  3  .  [3  =  1728. 

Ex.  3.  In  how  many  different  ways  can  the  letters  of  the 
word  volume  be  arranged  so  that  the  vowels  occupy  the  even 
places  ? 

The  vowels  can  be  placed  respectively  in  3,  2,  and  1  ways.  Similarly, 
the  consonants  can  be  placed  in  3,  2,  and  1  ways. 

Hence  the  required  number  is  3  •  2  •  1  •  3  •  2  •  1  =  36. 

Ex.  4.  How  many  three-lettered  words  can  be  made  from 
the  letters  a,  b,  c,  d,  and  e,  if  repetitions  are  allowed  ? 

The  first  place  can  be  filled  by  5  different  letters  ;  the  second  place  in  5 
different  ways  ;  and  the  third  place  in  5  different  ways.  The  total  number 
of  arrangements  is,  therefore, 

5.5.5  =  125. 
EXERCISE  149 

1.  In  how  many  different  ways  can  5  red  balls  and  3  black 
balls  be  arranged  in  a  row  ? 

2.  How  many  different  permutations  can  be  formed  from 
all  the  letters  of  the  word  algebra  ? 

3.  In  how  many  different  ways  can  the  letters  of  the  word 
exponent  be  arranged  ? 

4.  In  how  many  ways  can  the  letters  of  the  word  Mississippi 
be  arranged  ? 

5.  Find  the  number  of  permutations  that  can  be  made  from 
all  the  letters  of  the  word  parallelopiped. 

6.  How  many  different  permutations  can  be  made  from  all 
the  letters  of  the  word  trigonometry  ? 

7.  How  many  different  numbers  of  six  figures  can  be  formed 
from  the  digits,  1,  1,  1,  2,  2,  3  ? 

8.  In  how  many  different  ways  can  4  quarters,  2  dollars, 
and  3  dimes  be  arranged  in  a  line  ? 


394  ADVANCED  ALGEBRA 

9.   In  how  many  different  ways  can  3  white  and  4  blue 
flags  be  displayed  at  a  time,  one  above  the  other? 

10.  How  many  permutations  can  be  made  from  all  the  letters 
of  the  word  electricity  ? 

11.  How  many  different  permutations  can  be  made  from  the 
letters  of  the  word  radium  so  that  the  vowels  occupy  the  even 
places  ? 

12.  A  stage  accommodates  5  passengers  on  each  side.  In 
how  many  different  ways  can  6  persons  be  seated  if  2  of  them 
always  take  their  seats  on  one  side,  and  a  third  on  the  other 
side  ? 

13.  On  a  railroad  there  are  20  stations.  How  many  different 
tickets  are  required  to  connect  every  station  with  every  other 
one? 

14.  How  many  different  arrangements  beginning  with  the 
letter./  and  ending  with  a  consonant  can  be  formed  out  of  the 
letters  of  the  word  factor  ? 

15.  How  many  different  words  of  three  letters  can  be  formed 
out  of  the  letters  a,  b,  c,  e,  u,  when  the  letters  may  be  repeated  ? 

16.  How  many  numbers  of  five  figures  can  be  formed  with 
the  digits  1,  2,  3,  4,  if  the  digits  may  be  repeated  ? 

17.  A  railway  signal  has  3  different  arms,  and  each  arm 
may  be  placed  in  4  different  positions.  Find  the  total  number 
of  signals  that  can  be  made. 

COMBINATIONS 

452.  The  combinations  of  n  things  taken  r  at  a  time  are  the 
different  groups  of  r  things  that  can 'be  selected  from  the  n 
things  without  regard  to  their  order. 

Thus,  the  combinations  of  the  letters  a,  b,  c,  taken  two  at  a 
time  are  ab,  ac,  be.  The  arrangements  ab  and  ba  represent 
different  permutations,  but  the  same  combination. 


PERMUTATIONS  AND   COMBINATIONS  395 

Ex.  Write  all  combinations  that  can  be  formed  from  the 
letters  a,  b,  c,  d,  e,  taken  three  at  a  time. 

,  Write  first  all  combinations  containing  a.  Of  these,  take  first  all 
which  contain  a&,  etc.  In  this  manner  we  obtain  ahc,  abd,  abe,  acd,  ace, 
ade,  bed,  bee,  bde,  cde. 

453.  To  find  the  number  of  combinations  of  n  different  things 
taken  r  at  a  time. 

Let  X  denote  the  required  number  of  combinations.  Since 
each  of  the  combinations  contains  r  different  things,  we  could 
arrange  these  things  in  \r  different  ways,  or  each  combination 
would  produce  |_r  permutations.  Hence  the  total  number  of 
permutations  that  could  be  formed  would  be  x  -  \r.  But,  on 
the  other  hand,  this  number  represents  the  total  number  of 
permutations  that  can  be  formed  from  the  n  things  taken  r  at  a 
time,  or  "P^. 

Hence  ic -[r  =  "P^  =  n(7i  — 1)  •-.  (w  — r +  1). 

Therefore  ^^n  (n -1)  ...  (n-r  +  1). 

[r 

454.  This  number  was  represented  by  the  symbol  "(7,.  (§  374). 
Hence  we  may  represent  the  number  of  combinations  of  n 
things  taken  r  at  a  time  by  "(7,.,  and 

.(,   ._/!(/. -!)(/. -2)- (/.-/•  +  !)  ^jj 

455.  If  we  multiply  numerator  and  denominator  of  this 
formula  by  \n  —  r  (§  377),  we  obtain 

"a  =  ,— i^-  (2) 

\r\n  —  r  ^ 

Hence  (§  377) 

"0,  =  "0„_,.  (3) 


396  ADVANCED  ALGEBRA 

Note.     In  formula  (2),  if  n  =  r, 

\n 

Bilt  ""Cn  =  1,  hence  [0  =  1. 

Similarly,  «Co  =  "O^  =  1. 

Ex.  1.  In  how  many  different  ways  can  a  baseball  nine  be 
selected  from  11  candidates  ? 

This  is  evidently  the  number  of  combinations  of  11  elements  taken  9 
at  a  time. 

But  11(79  =11^2  =  ^^^-^  =  56. 

Ex.  2.  Out  of  the  first  10  letters  of  the  alphabet  how  many 
different  selections  of  4  letters  may  be  made  so  as  to  include 
the  letter  a  ? 

The  a  can  be  taken  only  one  way  ;  the  remaining  letters  can  be 
selected  in  i^Oa  different  ways.     Hence  the  total  number  of  selections  is 

10(7^  10- 9 -8^X20. 
1.2.3 

Ex.  3.  Out  of  5  Republicans  and  6  Democrats,  how  many 
different  committees  can  be  formed,  each  consisting  of  3  Repub- 
licans and  4  Democrats  ? 

The  number  of  ways  in  which  the  Republicans  can  be  selected  is  ^Cs, 
while  the  Democrats  can  be  chosen  in  ^d  different  ways.  Since  each  of 
the  first  group  can  be  combined  with  each  of  the  second  group,  the  total 
number  of  selections  is 

6C3  X  604  =  ^Cs  X  6(72  =  —  X  ^  =  150. 

Ex.  4.  How  many  words,  each  containing  2  consonants  and 
3  vowels,  can  be  formed  from  6  consonants  and  7  vowels  ? 

The  consonants  can  be  chosen  in  ^Ci^  the  vowels  in  ''Czi  different  ways. 

Hence  ^C^y.'^Cz  different  sets  of  letters  may  be  selected.     The  letters  of 

each  selection  can  be  arranged  in  ^P^  different  ways.    Hence  the  required 

number  is 

6C2  X  7(73  X  sPs  =  15  X  35  X  120  =  63,000. 


PERMUTATIONS  AND  COMBINATIONS  397 

Ex.  5.  How  many  different  sums  of  money  can  be  made  up 
by  combining  any  number  of  the  following  5  coins :  a  cent,  a 
dime,  a  half-dollar,  a  dollar,  and  a  five-dollar  piece  ? 

The  cent  may  be  disposed  of  in  2  different  ways,  i.e.  it  may  be  either 
included  or  not  included  in  the  sum.  Similarly,  each  coin  can  be  disposed 
of  in  2  different  ways,  hence  all  can  be  disposed  of  in  2x2x2x2x2 
=  2^  different  ways.  But  among  these  is  one  case  in  which  none  of  the 
coins  is  taken,  a  case  which  evidently  has  to  be  excluded.  Hence  the 
required  number  is 

25-1,  or  31. 

EXERCISE   160 

1.  Write  all  combinations  of  the  letters  a,  6,  c,  d,  e,  taken 

two  at  a  time. 

2.  Write  all  combinations  of  the  letters  of  the  word  surdj 
taken  three  at  a  time. 

3.  Write  all  combinations  of  the  letters  of  the  word  ratio, 
taken  three  at  a  time. 

4.  Find  the  value  of  ''C„  ''Go,  'O^. 

5.  If  "6^2  =  15,  find  n. 

6.  If  "0„_2  =  10,  find  n. 

7.  How  many  different  combinations  of  three  letters  can 
be  formed  from  the  26  letters  of  the  alphabet  ? 

8.  How  many  different  selections   of  20   letters   may  be 
formed  from  23  letters? 

9.  In  how  many  different  ways  can  a  committee  of  3  be 
selected  from  a  class  of  21  students  ? 

10.  How  many  different  straight  lines  are  determined  by  12 
points,  no  three  of  which  are  in  a  straight  line  ? 

11.  How  many  diagonals  can  be  drawn  in  a  decagon  ? 

12.  How  many  different  planes  are  determined  by  9  points, 
no  four  of  which  lie  in  a  plane  ? 


398  ADVANCED  ALGEBRA 

13.  How  many  different  triangles  are  determined  by  11 
points,  no  three  of  which  lie  in  the  same  straight  line  ? 

14.  Find  the  greatest  number  of  points  in  which  6  straight 
lines  may  intersect. 

15.  In  how  many  different  ways  can  a  guard  of  5  men  be 
formed  from  a  detachment  of  10  soldiers  ? 

16.  In  how  many  different  ways  can  4  letters  be  selected 
from  the  letters  of  the  word  equation^  if  each  selection  has  to 
contain  the  letter  e  ? 

17.  From  4  teachers  and  20  students,  how  many  different 
committees  can  be  selected,  each  consisting  of  2  teachers  and 
3  students  ? 

18.  How  many  words  can  be  formed  by  selecting  2  conso- 
nants and  3  vowels  from  5  vowels  and  20  consonants  ? 

19.  From  a  detachment  of  20  soldiers,  in  how  many  differ- 
ent ways  can  a  guard  of  6  be  selected,  when  2  particular  men 
are  always  excluded? 

20.  If ''04  =  "(7u,  findw. 

EXERCISE   151 
MISCELLANEOUS    EXAMPLES 

1.  If  the  number  of  permutations  of  n  things,  taken  3  at  a 
time,  is  equal  to  12  times  the  number  of  combinations  of  n 
things  taken  two  at  a  time,  find  n, 

2.  From  20  soldiers  and  10  sailors,  how  many  different  par- 
ties of  3  soldiers  and  2  sailors  can  be  formed  ? 

3.  If  "04  =  "(76,  find^Oy. 

4.  How  many  different  numbers  greater  than  52,000  can  be 
formed  from  the  figures  1,  2,  3,  4,  5  ? 


PERMUTATIONS  AND   COMBINATIONS  399 

5.  How  many  different  significant  numbers  can  be  formed 
with  the  digits  0,  1,  2,  3,  if  each  figure  may  be  repeated  ? 

6.  A  boat's  crew  consists  of  8  men,  of  whom  2  cati  row 
only  on  the  port  side  and  1  only  on  the  starboard  side.  Find 
the  number  of  ways  in  which  the  crew  can  be  arranged. 

7.  In  how  many  different  ways  can  the  word  algebra  be 
read  in  the  following  arrangement  of  letters,  if  we  commence 
at  the  first  letter  of  the  first  line  and  advance  from  letter  to 
letter  either  by  a  downward  step  or  a  step  to  the  right  ? 

algebra 

1   g  e   b   r  a 

g  e  b  r   a 

e   b   r  a 

bra 

r  a 

a 

8.  How  many  signals  can  be  made  with  8  different  flags,  if 
5  of  them  are  displayed  at  a  time,  one  above  another  ? 

9  How  many  signals  can  be  made  with  4  different  flags,  if 
any  number  of  them  may  be  displayed  at  a  time,  one  above 
another  ? 

10.  In  how  many  different  ways  can  3  dollars,  2  half-dollars, 
and  4  dimes  be  placed  in  a  circle  ? 

11.  How  many  arrangements  commencing  and  ending  with 
a  consonant  can  be  made  from  the  letters  a,  6,  c,  d,  e  ? 

12.  How  many  numbers  between  4000  and  6000  can  be 
made  with  the  digits  4,  5,  7,  8  ? 

13.  How  many  different  sums  can  be  made  up  by  combining 
any  number  of  the  following  coins :  a  cent,  a  nickel,  a  dime,  a 
quarter,  a  half-dollar,  and  a  dollar  ? 

14.  Find  w,  if  "P4  =  6"P3. 


400  ADVANCED  ALGEBRA 

15.  From  6  teachers,  10  boys,  and  10  girls,  how  many  com- 
mittees, each  consisting  of  2  teachers,  2  boys,  and  2  girls,  can 
be  selected  ? 

16.  How  many  different  throws  can  be  made  with  3  dice  ? 

17.  What  is  the  greatest  number  of  points  in  which  8 
straight  lines  can  intersect? 

18.  In  how  many  different  ways  can  7  presents  be  given  to  2 
children  so  that  one  receives' 3,  the  other  4,  presents  ? 

19.  How  many  different  numbers  of  5  figures  can  be  formed 
from  the  digits  1,  2,  3,  4,  5,  6,  7,  8,  so  that  the  first,  third,  and 
fifth  digits  are  odd,  if  no  digit  is  repeated  ? 


CHAPTER   XXVII 
DETERMINANTS 
DETERMINANTS  OF  THE  SECOND  ORDER 
456,   Consider  the  simultaneous  equations 

a^  +  622/  =  ^2' 
Solving,  we  obtain 

Oibz  —  ac^i   ^     Oi^a  — «A 
The  common  denominator  may  be  written  in  the  form 


This  symbol  is  called  a  determinant,  and  it  signifies  the 
difference  of  the  cross  products  of  the  four  quantities  involved. 


(1) 

(2) 
(3) 


Thus, 


and 


(h 

h 

a^ 

b2 

4 

5 

2 

3 

=  ai&2  —  «2&i? 


=  4.3-2.5  =  2. 


w 


457.  The  quantities  ai,  61,  az,  62  in  the  determinant 
are  the  elements  of  the  determinant. 
2d  401 


^2       ^2 


402 


ADVANCED  ALGEBRA 


458.  A  row  is  a  horizontal  line  of  elements,  a  column  a 
vertical  line  of  elements. 

The  rows  in  (4)  are  a\bi  and  a2&2. 

The  columns  are     ^  and  /• 
a^  02, 

459.  The  expression  aih^  —  a^hi  is  said  to  be  the  expansion  of 
the  determinant. 

The  terms  of  a  determinant  are  the  terms  of  its  expansion ; 
as,  aj)2' 

460.  The  order  of  a  determinant  is  the  number  of  terms  in 
a  row  or  a  column. 


is  a  determinant  of  the  second  order. 


The  principal  diagonal  is  composed  of  the  elements  from 
the  upper  left-hand  corner  to  the  lower  right-hand  corner  j 
as,  a-J)2- 

The  line  a^i  is  called  the  secondary  diagonal. 


461.   The  roots  of  equations  (1)  and  (2)  may  be  written 


x  = 


Ci       &1 

tti       Ci 

Ca     62 

,  y  = 

a2    c. 

«!        &1 

ai    61 

0^2       h 

as    62 

J.e.  the  common  denominator  is  the  determinant  composed 
of  the  coefficients  of  x  and  y. 

The  numerator  of  the  value  of  x  may  be  formed  from  the 
denominator  by  substituting  Cj  and  C2  in  place  of  the  corre- 
sponding coefficients  of  x  (i.e.  ai,  a^. 

The  numerator  of  y  may  be  formed  from  the  denominator 
by  substituting  Cj  and  Cg  in  place  of  the  coefficients  of  y. 


Ex.    Solve  the  system 


DETEBMINA  N  T8 


408 


X 


2x  +  3y  =  T, 


32 
16 


16 
16 


6 

-2 

7 

3 

4 

-2 

2_. 

3 

2. 


4  '^     6 

2         7 


Expand : 


4     -2 
2         3 

EXERCISE  152 


1. 


2. 


3. 


4. 


-1     -2 
3     -5 


5. 


0 
-3 


Solve; 
11. 


12 


13 


■I 


3     X 
2     2/ 


4a;-2/  =  13, 
5  a;— 2/  =  17. 

2x-2y  =  -2. 
x-\-2y  =  5, 


10. 


a    3a 

6     26 

a  4-  6     a  +  c 
a  — c     a  — 6 


a     —a" 

a    a; 
0    0 


^^     (2x-}-5y  =  17a, 
\Sx  —  7y=   5  a. 


x-\-3y  =  7. 

14.    i  „ 

I  a;  —  ■?/==  a  —  J. 


16. 


17. 


18. 


f  ma;  —  ny  =  0, 
[   cx  —  dy=:l. 

mx  -\-ny  =  py 
I  aa;  +  62/  =  c. 

(ax  —  by  —  c, 
[ax  +  by  =  d. 


404 


ADVANCED  ALGEBRA 


Prove  the  following  identities : 


19. 


20. 


ai 

h 

h,    i 

^.1 

ag 

h 

h    ag 

aj 

h 

ai    012 

a^ 

h 

h. 

&2 

21. 


22. 


23.   a:b  =  c:d,  ii 


mai     fej 
7na2    b^ 

a     b 
c     d 

a     b 
c     d 


=  m\ 


Oj     bi 


a.2    62 
=  0,  if  a\b  =  c'.d. 


=  0. 


Express  as  determinants : 
24.  4.5-3.2.      25.  6.7-3(-2).      26.  22-17.      27..  a?-b\ 


DETERMINANTS   OF  THE   THIRD  ORDER 

462.  In  any  term  whose  literal  factors  follow  in  natural 
order,  the  occurrence  of  a  larger  subscript  before  a  smaller  one 
is  called  an  inversion. 

Thus,  aibzC2  contains  one  inversion,  viz.  &3C2. 

azhiCi  contains  two  inversions,  viz.  a^bu  a^c^- 
asbiCi  contains  three  inversions,  viz.  aab2,  aaCi,  &2C1. 


463.    If  we  solve  the  three  simultaneous  equations 

(a^x  4-  b^y  +  CjZ  =  d^, 
a>fc  +  boy  +  c^  =  d^y 
a^-\-b^  +  c^z  =  d^, 
we  obtain  for  x 

^  _  dib<fi^  —  d^bgC^  -h  d-ibsC^  —  d^b^c^  +  dg&iCg  —  d^^ 


(1) 

(2) 
(3) 

(4) 


The  denominator  is  usually  written  as  a  determinant  of  the 
third  order,  or 


0162^3  —  (h^sC2  +  agftaCi  —  a-ibiC^  +  0.36102  —  a^b^Ci  = 


a, 

^ 

Ci 

a^ 

b. 

C2 

(I3 

bs 

C3 

(5) 


DETERMINANTS 


406 


464.  A  determinant  of  the  third  order  is  equal  to  the  sum  of 
all  the  products  that  can  be  formed  by  taking  one  element,  and 
only  one,  from  each  column  and  from, ea/ih  roiv,  considering  the 

.  sign  of  each  product  'positive  if  it  contains  an  even  number  of 
inversions,  and  negative  if  it  contains  an  odd  number  of 
.inversions. 

Thus,  aihzC2  contains  one  inversion,  hence  it  is  negative  ; 

a^bzCi  contains  two  inversions,  hence  it  is  positive ; 

a^bzCi  contains  three  inversions,  hence  it  is  negative. 

465.  The  annexed  figure  offers  a  +  +  + 
convenient  method  for  expanding  a 
determinant  of  the  third  order,  but 
it  should  be  borne  in  mind  that  this 
method  does  not  apply  to  determinants 
of  a  higher  order  than  the  third. 

466.  By  grouping  the  terms  of  the  expansion  according  to 
ax,  ctg,  and  a^,  and  denoting  the  determinant  (5)   by   D,  we 

D  =  aiib^c^  —  b^c^  —  a2{biC3  —  b^Ci)  +  03(6102  —  Vi), 

ai    bi 


or 


«3       63 


Ci 

62    C2 

h 

Ci 

bi     Ci 

C2 

=  ai 

-a^ 

+  ^3 

bs     <h 

h 

C3 

62     C2 

C3 

(6) 


The  coefficients  of  ai,  —02,  and  a^  are  called  the  minors  of 
these  elements. 


467.  The  minor  of  any  element  is  the  determinant  obtained 
by  erasing  the  row  and  the  column  which  contains  the 
element. 

Thus  the  minor  of  h\  is  obtained  by  erasing  the  first  row  and  tho 
second  column,  i.e. 


^1     } 

♦      r. 

-C/l         i/ 

1     '^l 

02      C2 

ai    I 

2      C2 

or 

08      Cq 

08      I 

8      C8 

406 


ADVANCED  ALGEBBA 


hi 

Cl 

bs 

cz  ' 

ax 

bi 

az 

&2 

468.    If  an  element  occurs  in  the  mth  row  and  in  the  nth 
column,  its  minor  multiplied  by  (-1)"*+"  is  called  its  co-factor. 


Thus,  the  co-factor  of  a2  is  — 
Thus,  the  co-factor  of  cs  is  + 


469.    The  co-factors  of  ai,  a^^  a^,  b^,  etc.,  are  usually  denoted 
by  Aj  a,  a,  Si,  etc. 


Thus,  B2 


470.    Using  this  notation,  (6)  may  be  written 


«1      Cl 

;  C2  =  - 

«i     bi 

^3      C3 

as    bs 

D  = 


ffl 

b. 

Cl 

fl2 

b. 

C2 

as 

bs 

Cs 

=  aJi  +  02^2  +  fls^s- 


471.    By  grouping  the  terms  of  the  expansion  according  to 
the  elements  of  the  second  row  {i.e.  a2,  h^,  C2),  we  have 


D  =  —  a2 


oi 


bi 

Cl 

+  &2 

tti 

Cl 

-C2 

ai 

^'i 

h 

C3 

ag 

C3 

«3 

h 

D  =  02^2  +  62^2  +  C2^2- 


Similarly  for  any  row  or  column.    Hence  we  have  in  general : 

472.  A  determinant  of  the  third  order  is  equal  to  the  sum  of 
the  products  obtained  by  midtiplying  the  elements  of  any  row  or 
column  by  their  respective  co-factors. 

473.  The  evaluating  of  determinants  by  means  of  co-factors 
is  usually  more  convenient  than  by  direct  expansion. 


DETEBMINANTS 

Ex.  1.    Find  the  numerical  value  of 
4    3     2 


407 


2    3 
1    2 


Expanding  in  terms  of  the  elements  of  the  first  row,  we  have 


2    3 
1     2 


1     3 

4    2 


+  2 


1     2 
4     1 


=  4.1-3.  (-10) +2.  (-7)  =20. 

Ex.  2.    Eind  the  numerical  value  of 
2     -1     3 


1  0 

2  1 


Since  the  third  column  contains  a  zero,  expand  in  terms  of  the  elements 
of  the  third  column. 


4 

1 

-1 

+  1 

2 

-  1 

2 

-2, 

2 

-2 

4 

1 

=  3.(_10)-0  +  6=-24. 


Ex.  3.    Expand 


b-\-c 
b 
c 


a  a 

c-f-a         b 
c        a-f  6 


Expanding  in  terms  of  the  elements  of  the  first  row,  we  have 


(6  +  c) 


c  +  a        h 
c        a  +  6 


+  a 


h    c-\-  a 


b        b 
c    a  -\-  b 

=  (6  +  c)  (a2  -f  a&  +  ac)  -  a(ab  +  62  _  6c)  +  a(bc  -  c^  -  ac) 

=  a^b  +  ab^  +  a6c  +  a'^c  +  a6c  +  ac"^  —  a^b  -  ab^  +  abc  +  abc  —  ac'^—aH 

=  4  abc. 


408 


ADVANCED  ALGEBRA 


EXERCISE  153 

1.   Determine  the  number  of  inversions  in  ai^gCgd^,  ap^CT^d^^ 
afiiC^d2,  and  a^h^p^d^^^. 

Oil      W     Cl 

t*2       ©2       Cg 


2 .  In  the  expansion  of 

3.  Expand 


«3     &a 


what  is  the  sign  of  a^-fiz. 


mi    rii 

i?i 

m2    ^2 

^^2 

mg        713 

i^a 

4.  In  the  determinant  of  Ex.  3,  what  is  the  minor  (a)  of 
W2?  (&)ofi)2?  (c)  of  mg? 

5.  In  the  same  determinant  what  is  the  co-factor  of  (a)  %? 
(&)  of  m^  ?  (c)  of  7i3  ? 

12        3 
4     5-1 


6.    In  the  expansion  of 


,  find  the  coefficient  of  x. 


7.  In  the  same  determinant,  what  is  the  coefficient  of  ?/  ? 

8.  Expand 


2  1 

3  2 

1     2 


Find  the  value  of  the  following  determinants 


12. 


13. 


1 

2 

3 

4 

2 

1 

. 

2 

4 

6 

1 

a 

a 

a 

3 

2 

. 

a 

1 

a 

4 

9 

-3 

2 

1 

-1 

1 

7 

2 

10. 


2     0 


14. 


15. 


4 

0    0 

11. 

7 

2    3  . 

8 

2    4 

+  x        1 

1 

1        1+2/ 

1 

1           1 

1  +  ^ 

0     4     5 

4  0     2 

5  2    0 


DETERMINANT^ 


409 


Solve  the  equations : 

A:  X  X 

16.     2-1         3 
1         2-1 


17. 


2  3        1 

3-12 

1  +  aj    1  —  x    X 


Express  as  determinants  of  the  third  order : 


18.  nil 


19.  a(n—p)  —  'b{m—p)-\'c(m--n). 


h 

1 

hr 

1 

^ 

1 

h 

1 

-m2 

h 

1 

4-WI3 

h 

1 

=  0. 


DETERMINANTS  OF  THE  FOURTH  ORDER 

474.  The  principles  derived  for  determinants  of  the  third 
order  are  also  applicable  to  determinants  of  the  fourth  and 
higher  orders. 

475.  Thus  the  symbol 


ai 

\ 

Ci 

d. 

a2 

h 

C2 

d. 

as 

h 

C3 

ds 

a. 

h 

C4, 

d. 

is  a  determinant  of  the  fourth  order,  and  it  is  equal  to  the  sum  of 
all  the  products  that  can  be  formed  by  taking  one,  and  only  one, 
element  from  each  column  and  from  each  row,  considering  the 
sign  of  each  product  positive  if  it  contains  an  even  number  of 
inversions,  negative  if  it  contains  an  odd  number  of  inversions. 

476.  Since  the  terms  of  the  expansion  contain  the  same 
literal  factors,  but  differ  in  the  arrangement  of  the  subscripts, 
we  may  obtain  all  terms  by  forming  all  permutations  of  the 
subscripts  1,  2,  3,  and  4. 

Hence  the  total  number  of  terms  is  *Pi,  i.e.  [4,  or  24. 


410 


ADVANCED  ALGEBRA 


477.  If  all  the  terms  containing  ai  are  combined,  the  coeffi- 
cient of  tti  cannot  contain  any  elements  of  the  first  row  or  the 
first  column,  but  it  must  consist  of  the  sum  of  all  products  that 
can  be  formed  by  taking  one  element,  and  only  one,  from  the 
last  three  columns  and  last  three  rows.  The  signs  of  these 
products  are  positive  or  negative  according  as  the  number  of 
inversions  is  even  or  odd. 

Hence  the  coefficient  of  Oj  is  the  minor  of  a^  i.e. 


b, 

Cg         ^2 

h         C3         dg 

d^          C4          ^4 

If  the  terms  containing  ofg  were  collected,  we  should  obtain  in 

a  similar  manner 

6i     Ci     d] 

-as 

63         C3         dg 

. 

64         C4 

d. 

Collecting  the  terms  containing  a^  and  a^,  we  have 
«!  61  Ci  dy 

0,2    O2    ^2    1X2 

a.}  h  C3  d^ 
a^  64  C4  ^4 

Evidently  the  coefficient  of  every  element  is  the  co-factor  of 
the  element.  Hence,  using  the  same  notation  as  in  the  preced- 
ing chapter,  we  have 

D  =  fli^i  +  02^2  +  03^3  +  a  J  A- 


h  C2  d^ 

h,    Ci  'di 

61   Ci  di 

5i  ot  (^1 

=  «! 

63  C3  d^ 

-^2 

63    C3    ^3 

+  ^3 

62    C2    (^2 

-a4 

62    C2    C?2  . 

64  C4  d. 

64    C4    (^4 

&4    C4    0^4 

63    C3    ^3 

478.   In  a  similar  manner  the  determinant  may  be  expanded 
in  terms  of  the  elements  of  any  row  or  any  column. 

E.g.  the  expansion  in  terms  of  the  elements  of  the  third  row  is 

03^3  +  &3-B3  +  C3C3  +  dzDz. 


DETERMINANTS 


411 


479.  If  all  elements  in  a  row  or  a  column  are  zero,  the  deter- 
minant equals  zero. 

For  expanding  in  terms  of  the  elements  of  that  row  or  col- 
umn, each  term  becomes  zero. 

480.  If  all  the  elements  hut  one  in  a  row  or  a  column  are 
equal  to  zero,  the  determinant  is  equal  to  the  product  of  the 
element  and  its  co-factor,  and  hence  reduces  to  a  determinant  of 
the  next  lower  order ;  e.g. 


ai    bi    ci    0 

ai    &i    Ci 

a2    h-i    G2    0 

=  -d 

az    &2    C2 

az    &3    C3    d 

a4    &4    C4 

054      64      C4      0 

Ex.  1.    Find  the  numerical  value  of 

4 
1 

2 
3 

Expanding  in  terms  of  the  elements  of  the  second  row, 


2 

1 

1 

0 

2 

3 

1 

2 

0 

2 

0 

1 

2     1     1 

1     2     0 

+  0-2 

2     0     1 

2     1 

1  0 

2  1 


+  3 


4 
2 

3         2 


2     1 
1     2 


=.-!(+ l)+0-2(-l)  +  3(+3) 
=  -14.2  +  9  =  10. 


Ex.  2.    Find  the  coefficient  of  x  in  the  expansion  of 


a  X 

2  5 

2  -11 

4  27 


2/  n 

1  3 
0  2 

2  1 


412 

The  coefficient  is 


ADVANCED  ALGEBRA 


2     13 

2    2 

2     0     2 

=  + 

4     1 

-0  +  2 

4    2     1 

_  6  -  4  =  -  10. 


2    3 

2     2 


EXERCISE   154 

4         2x1 

1.  In  the  expansion  of  the  determinant  find 

the  coefficient  of  a;.  ~       ^ 

1        a    2    6 

2.  In  the  same  determinant  find  the  coefficient  of  y. 

3.  In  the  same  determinant  find  the  coefficient  of  z. 

4.  In  the  same  determinant  find  the  coefficient  of  a. 


Find  the  value  of  the  following  determinants : 


6. 


4 

2 

-1 

17 

2 

9 

0      6 

1 

1 

0      2 

2 

3 

0      4 

4 

5 

19     7 

0 

0 

0    0 

2 

1 

7     2 

* 

4 

5 

6     7 

a 

0 

0    0 

0 

a 

0     0 

0 

0 

a    0 

0 

0 

0     a 

8. 


10. 


a     0 

0 

133 

2     0 

a 

77 

0     0 

0 

a 

* 

y   €L 

a; 

77 

2 

1 

2 

4 

0 

0 

3 

0 

4 

9 

-3 

2 

-2 

1 

0 

2 

0     1 

1 

1 

1     0 

1 

1 

1     1 

0 

1 

1     1 

1 

0 

DETERMINANTS 


413 


11.   Reduce  to  a  determinant  of  the  third  order 
a    h     0     c 


d    e 
9    h 


0    / 

i     k 

I     m    0    p 

12.   Reduce  to  a  determinant  of  the  second  order 
a     0     0    0 
X     y     b     c 


0    n 
0     r 


X 


GENER-AL  PROPERTIES  OF  DETERMINANTS 

481.    The  value  of  a  determinant  is  not  altered  when  the  rows 
are  changed  to  columns  and  the  columns  to  rows. 


.e.  the  proposition  is  true  for 


Obviously 

as 

b. 

= 

ai 

0^2 
62 

j 

determinants  of  the  second  order. 

«!    6i    Ci 

Let           D  = 

aa     62     C2 

and 

aa 

&3 

C3 

ai 

aa 

^3 

^^1 

h 

&3 

Cl 

C2 

Cs 

Expanding  D  in  terms  of  the  elements  of  the  first  column, 
and  D'  in  terms'  of  the  elements  of  the  first  row,  we  have 

D  =  a^ 

D'  =  a, 

Hence     D  =  D'. 

Similarly  for  determinants  of  higher  orders. 


&2 

C2 

h. 

Ci 

h 

Cl 

-^2     ^ 

+  013 

&3 

C3 

h 

Cs 

b. 

C2 

h 

h 

-a^ 

&i 

&3 

+  a^ 

h 

&2 

C2 

C3 

Ci 

C3 

Ci 

C2 

414 


ADVANCED  ALGEBRA 


482.  Any  theorem  ivhich  is  true  for  the  columns  of  a  determi- 
nant is  true  for  its  rows,  and  vice  versa. 

483.  The  interchange  of  any  two  adjacent  columns  (or  rows) 
of  a  determinant  changes  the  sign  of  the  determinant. 


E.g.  let  D 


tti  hi  Ci  di 

a^  ^2  C2  ^2 

<^3  ^3  ^3  ^3 

a^  64  C4  C?4 


and  i)'  = 


Ci 

\ 

di 

Cj 

\ 

d, 

C3 

63 

(?3 

<:^ 

6, 

d. 

The  minor  of  h^  in  D  is  evidently  equal  to  the  minor  of  h^  in 
D\  but  their  respective  co-factors  have  opposite  signs.  Hence, 
denoting  the  co-factor  of  h^  in  D  by  By,  the  co-factor  of  hy  in  Z)' 


is  — -Bij  etc. 

Hence 

Z)  =  \Bi  -f  62  A  +  ^3^3  +  \^,^ 

and 

D'  =  -  h,Bi  -  hA  -  bA  -  hA- 

Therefore 

D  =  -D\ 

484.  The  interchange  of  any  two  columns  (or  rows)  of  a  deter- 
minant changes  the  sign  of  the  determinant. 

For  the  interchange  of  any  two  columns  can  be  effected  by 
an  odd  number  of  successive  changes  of  adjacent  columns. 
E.g.  to  interchange  the  1st  and  4th  columns  make  three  suc- 
cessive interchanges  of  adjacent  columns,  viz.  1st  and  2d,  2d 
and  3d,  3d  and  4th.  The  former  4th  column  is  now  the  3d, 
hence  two  successive  changes  will  carry  it  to  the  first  place. 
Or  the  total  number  of  exchanges  is  5,  and  (— 1)''  =  — 1. 
Similarly,  if  m  interchanges  are  necessary  to  bring  a  particular 
column  in  place  of  any  other  one,  m  —  1  will  bring  the  second 
one  in  place  of  the  hrst,  or  the  total  number  of  interchanges  is 
2  m  —  1 ;  i.e.  an  odd  number. 


E.g. 


ao 

a 

1 

1 

a 

a2 

&2 

h 

1 

=  - 

1 

b 

h2 

C2 

c 

1 

1 

c 

C2 

DETERMINANTS 


415 


485.    If  two  columns  {or  roics)  of  a  determinant  are  identical, 
the  determiyiant  vanishes;  i.e.  it  is  equal  to  zero. 


Let 


D 


a. 

6i 

tti 

^1 

"2 

h 

as 

C2 

a. 

h 

ag 

C3 

64    a4 


By  the  preceding  paragraph  the  interchange  of  the  two 
identical  columns  changes  the  sign  of  the  determinant ;  i.e.  the 
resultant  determinant  equals  —  D. 

But,  on  the  other  hand,  an  interchange  of  two  identical 
columns  does  not  change  the  determinant. 


Hence 

D  = 

-D,  or 

2D  =  0 

Therefore 

D  =  0. 

6    a;    6 

E.g. 

7    y    7 
a    z    a 

=  0. 

486.  If  the  elements  of  any  column  (or  row)  he  multiplied  by 
the  cof actors  of  the  coi^responding  elements  of  any  other  column 
(or  row),  the  sum  of  the  products  vanishes. 

For  such  a  sum  represents  a  determinant  with  two  identical 
rows  or  columns. 

61      &i      Ci 

E.g.  biAi  +  62  A  +  hAi  = 


62    62 


=  0. 


Similarly,    02^1  +  h^i  +  c^Ci  =  0. 


487.    If  all  elements  in  any  column  are  multiplied  by  any  factor , 
the  determinant  is  multiplied  by  that  factor. 


416 


ADVANCED  ALGEBRA 


For 


mai 

h 

Cl 

ma2 

h 

C2 

mos, 

h 

C3 

=  mOiAi  +  ma2^2  +  'ina^A^ 


=  m  (aiAi  +  ^2^42  +  a^A^) 


=  m 


ai 

^ 

Ci 

^2 

2>2 

C2 

as 

?>3 

C3 

This  principle  is  also  true  for  a  division,  for  a  division  by  m 
is  equivalent  to  a  multiplication  by  — 


Thus, 


mai 

ai 

aa 

mh\ 

&i 

62 

=  m 

mci 

Cl 

C2 

«i 

«i 

a2 

&i 

&i 

&2 

Cl 

Cl 

C2 

Or 


33    30 

66     75 


33  .  15 


1  2 

2  5 


33  .  15  •  1  =  495. 


488.  If  sack  element  of  any  cohimn  {or  row)  of  a  determinant 
is  the  sum  of  two  quantities,  the  determinant  can  be  expressed  as 
the  sum  of  two  determinants  of  the  same  order. 


For 


ai  4-  di  bi 
a-2  +  ^2  62 
%  +  ^3     ^3 


=  (ai  +  C?i)A+K  +  C?2)^2+(a3  +  ^3)^3 


=  (Oi^l  4-  a2^2  +  a3^3)  4-  (^1^1  +  ^2  A  +  <^3^3) 


0^2        ^2 
^3       &3 


4- 


d,     61 

^2       &2 

(^3    b. 


^.g. 


4  +  1    4 

5  +  2    5 


4  41      11    4 

5  5r    2    6 


1  4 

2  6 


DETERMINANTS 


417 


489.  If  each  element  of  a  column  is  multiplied  by  the  same 
number,  and  the  products  be  added  to  {or  subtracted  from)  the 
corresponding  elements  of  another  column^  the  determinant  is  not 
altered. 

According  to  the  preceding  paragraph  we  have 


a.2  ±  mbi    62    C2 
%  ±  "i^h     h     C3 


Oi      61 

Ci 

a2    &2 

C2 

± 

as    h 

<^ 

mbi 

\ 

<h 

mbi 

b. 

Oi 

mb^ 

h 

C3 

But  the  last  determinant  equals 


b,     61 

Ci 

m 

'&2       ^2       ^2 
63       63       C3 

and  hence  vanisli 

Ex.     Evaluate 

4244     4245 
4246     4247 

• 

The  determi 

nant  = 

4244 
2 

4245 
2 

= 

4244     1 
2      0 

=  -2. 


490.  Evaluation  of  determinants.  By  means  of  the  preceding 
proposition  all  elements  but  one  in  a  column  (or  row)  can  be 
made  equal  to  zero,  and  hence  the  determinant  can  be  reduced 
to  one  of  the  next  lower  order  (§  480).  In  many  cases,  how- 
ever, the  determinant,  before  reduction  to  a  lower  order,  should 
be  simplihed  as  follows : 

(1)  Eemove  factors  common  to  all  elements  in  a  row  or  a 
column. 

(2)  Diminish  the  absolute  values  of  the  elements  by  sub- 
tracting the  corresponding  elements  of  other  columns  (or  rows) 
or  multiples  of  these  elements. 

2e 


418 


ADVANCED  ALGEBRA 


Ex.  1.  Evaluate 


410  S6  51 
420  88  57 
300  60  120 


Remove  from  the  three  columns  respectively  the  factors  10,  2,  3,  and 
from  the  last  row^  the  factor  10.     The  result  is 


600 


41  43    17 

42  44     19 
3      3      4 


Subtracting  the  first  column  from  the  second,  we  have 
600 


41 

2 

17 

42 

2 

19 

3 

0 

4 

Subtracting  the  first  row  from  the  second, 
41     2     17 


600 


Ex.  2.   Evaluate 


10     2 
3    0     4 


-  600  .  2  . 


1     2 


=  - 

1200  ( 

:-2) 

=  2400. 

3 

2 

-2 

3 

4 

3 

7 

-2 

5 

1 

2 

3 

6 

2 

-3 

1 

From  the  first  row  subtract  the  third ;  to  the  second  add  twice  the 
fourth  ;  from  the  third  subtract  three  times  the  fourth.     The  result  is 


-2 

1 

-4  0 

16 
-13 

7 
-5 

1  0 
11  0 

,  which  reduces  to 

6 

2 

-3  1 

-2 

1 

-4 

16 

7 

1 

-13 

-5 

11 

DETERMINANTS 


419 


To  the  first  column  add  twice  the  second ;  to  the  third  column  add 
four  times  the  second : 


-1 


0 

1 

0 

30 

7 

29 

.     Thisr 

educi 

-23 

-5 

-9 

30 
-23 

29 
-9 

=  - 

-1 

7 
-23 

20 
-9 

=  -397. 


Ex.  3.   Evaluate 


—  a  +  b-{-c  2a  1 
a—b+c  2b  1 
a-f  b  —  c    2  c    1 


Adding  the  second  column  to  the  first,  the  determinant  becomes 


a  +  b-\-  c  2a  1 
a  +  6.+  c  26  1 
a-^h  +  c    2c    1 


=  2(a  +  6  +  c) 


1  a  1 
16  1 
1    c    1 


=  0. 


Ex.  4.   Evaluate  D=^ 


be 


ac 


ab 


Dividing  the  determinant  by  a6c,  and  multiplying  the  three  rows 
respectively  by  a,  6,  and  c,  we  have 


2>  = 


abc 


abc    1    a2 

1    1    a2 

abc    1     62 

= 

1      1      62 

a6c    2    c2 

1    2    c2 

1    0    a2 

1    a2 

1    0    62 

=  -1 

1    62 

1    1    c2 

=  a2  -  62. 


420 


ADVANCED  ALGEBRA 


491.  Factoring  of  a  determinant.  If  a  determinant  vanishes 
when  any  element  b  is  substituted  for  another  element  a,  then 
a  —  6  is  a  factor  of  the  determinant  (§  288). 


Ex.  1.   Factor  D  = 


1  a  a^ 
1  b  b' 
1     c     c2 


If  a  =  h,  the  resulting  determinant  will  contain  two  identical  rows, 
and  hence  will  vanish. 

Therefore  a  —  &  is  a  factor  of  D. 

For  similar  reasons  (&  —  c)  and  (c  —  a)  are  factors. 

Since  the  product  of  these  three  factors  is  of  the  same  degree  as  D, 
(a  —  b)(b  —  c){c  —  a)  and  D  can  differ  only  in  a  numerical  factor, 
e.g.  K. 

Therefore.  D  =  K{a  -  b){b  -  c)  (c  -  a). 

The  terra  obtained  from  the  principal  diagonal  in  D  is  bc^.  This  term 
must  be  equal  to  the  similar  term  of  the  right  member,  viz.  Kbd^. 


Hence 


Therefore 


bc^  =  Kbc^     Or  K=l, 


D  =  {a-b){b-c){G-a). 


a 

—  n 

a 

X 

b 

y 

X 

—  c 

y 

Ex.  2.   Factor  Z)  = 


If  «=?/,  or  6  =  —  c,  the  determinant  will  contain  two  identical 
columns,  or  rows,  and  hence  vanish. 

If  a  =  0,  Z)  is  reduced  to  a  determinant  of  the  second  order,  which  also 
vanishes. 

Since  D  is  of  the  third  degree,  we  have 

D  =  K{a-0){x-y){b  +  c). 
Equating  two  terms,  aby  =  —  Kaby,  i.e.  K  =  —l. 
Therefore  D  =  -  a(x-  y){h  +  c;. 


DETERMINANTS 


421 


EXERCISE  165 


Find  the  value  of  the  following  determinants 


1. 


67800  6781 
67820  6783 


a 

7 

a 

h 

X 

b 

c 

19 

c 

4     9      2 

2     0      2 
6     3-1 


9. 


3  11 
13  1 
111 

2  12 
12  1 
2    12 


1111      90 
9999    900 


6. 


6. 


10 

120 

15 

10 

130 

2 

10 

140 

9 

4:a  X  a 
4:b  y  b 
4  c    z     c 


5a    0 

20 

5b    2 

30 

5c    3 

10 

10. 


11. 


12. 


10 

4 

0 

20 

12 

6 

30 

8 

12 

12  2 
3-12 
6   11 


101  102  103 
104  105  106 
107  108  110 


13. 


1 

b 

a 

a 

1 

b 

b 

a 

1 

15. 


a-\-b   c      c 

lab 

a   b-\-c  a 

17. 

a    4    6 

b      b   c-\-a 

a    b    2 

14. 


0 

a'   b 

&2 

0    a^ 

a 

b'   0 

16. 


a 

b 

c 

d 

a 

b 

e 

d 

a 

18. 


a  2  a  -7 

b  5  b  2 

c  7  c  3 

d  9  d  4: 


422 


ADVANCE!)  ALGEBRA 


19. 


20. 


21. 


X  a  Sx  —11 

y  b  3y  -12 

z  c  3z  -13 

u  d  3u  —14 


4 

5 

6 

0 

4 

2 

1 

0 

2 

0 

1 

0 

7 

5 

11 

2 

12    3     1 

2  3     12 

3  12    3 
12    3    0 


22. 


23. 


12     3  4 

2  4    5  2 

3  12  1 
12    2  2 

4      3-1-2 

-1-2      2      1 

3      4-34 

12      3-1 


Resolve  into  factors 


24. 


25. 


a 

a 

a 

X 

X 

b 

b 

c 

d 

1 

1 

1 

1 

a 

a' 

1 

b 

W 

26. 


27. 


a& 

1 

c 

be 

1 

a 

ca 

1 

b 

.  Prove  the  following  identities : 


30. 


=  b(c-y)(x-y).      31. 


28. 


a 

a' 

a' 

b 

b' 

W 

c 

c' 

& 

d 

d' 

d' 

111 

1  1+a    1 

•     29. 

1      1     1  +  6 

a  X  a  a 

b  b  y  b 

c  c  c  z 

d  d  d  d 


1111 
1  1  +  a  1  1 
1  1  1  +  6  1 
1       1      1   1+c 


=  abc. 


32. 


x-\-  a  a;+2a  x  +  Sa 
x-{-2a  aj  +  3a  a;  +  4a 
aj  +  3a    aj  +  4a    aj  +  5a 


=  0. 


DETETtMINANTS 


423 


33. 


34. 


35. 


116     1 


all 
1  1  c 
111 


=  (a-l)(6-l)(c-l). 


1 

a 

a' 

1 

b 

b' 

1 

c 

c« 

(a  -  b)(b  —  c){c  —  a){a  +  6  +  c). 


b^  +  c^  ab  ca 
ab  c^-{-a^  be 
ca         be      a^-{-  b^ 


=  4.a^b^c'. 


Find  the  value  of  the  following  determinants 


36. 


(n-\-iy     (n+2)2     (n+Sy 
(n+2)2     (71+3)2     (71+4)2 

1+x     2         3         4 

1  2-\-x  3  4 
1  2  3+a;  4 
12         3      4+aj 


37. 


38. 


a^  be    ac  +  c^ 

a^  +  a6       6^        ac 
ba      b^+be    c" 


Simplify  the  following  expressions : 

39.  (ai  +  5i  +  Ci)(^i  +  A  +  Ci)  +  (a2  +  62  +  C2)(^2  +  JSg  +  Q  + 
(a3  +  63  +  C3)(^3+B3+ C3),  if  a,  6,  c  are  the  elements  of 
a  determinant  of  the  third  order,  and  A,  B,  C  their 
respective  co-factors. 


40. 


41. 


a  +  bi    a  +  d 

a  —  ci     a  —  bi 

i     i      5 

1      4:      2i 

,    if 

2    2      i 

if  i  =  V-i. 


iii=V^^. 


424 


ADVANCED  ALGEBRA 


SOLUTION  OF   LINEAR  SIMULTANEOUS  EQUATIONS 
AND  ELIMINATION 


492.   Consider  the  simultaneous  equations : 
azX  +  b^  +  C2Z  =  d2, 


Let 


D 


ai     61 

Ci 

02    62 

C2 

as    ^3 

C3 

(1) 

(2) 
(3) 

(4) 


Multiplying  the  first  three  equations  respectively  by  Ai,  A2, 
and  A^  (i.e.  the  co-factors  of  a^  Og,  Og),  we  have 

QiAjX  4-  61^1?/  +  Ci^i2  =  diAi,  (5) 

a2^2^  -I-  &2^22/  +  ^2^22!  =  C?2^2j  (6) 

ag^go:  +  63^3^  4-  CsA^z  =  (^3^3.  (7) 

Considering  that   61  J.i  +  feaA  +  b^As  =  0, 
and  Cj^li  +  C2^2  +  ^3^3  =  0, 

we  obtain  by  addition  of  (5),  (6),  and  (7), 

(ciiAi  +  a2^2  +  03^3)^  =  (^lAi  +  <^2^2  +  da^v 


Hence 


That  is, 


_  dl  A  +  ^^2^2  +  C^3-^8 
aiA-f-«2^2+a3^8' 


dl 

h 

Ci 

0^2 

h 

C2 

^3 

h 

C3 

tti 

h 

Ci 

(h 

62 

C2 

a. 

^^3 

C3 

DETERMINANTS 


425 


In  a  similar  manner,  we  obtain: 


y= 


a. 

d. 

Ci 

^2 

d. 

C2 

«3 

ds 

C3 

«! 

&1 

Ci 

a2 

&2 

C2 

(h 

h 

C2 

«1 

&i 

d. 

tta 

&2 

d, 

Og 

h 

ds 

«1 

h 

Ci 

tta 

h 

C2 

as 

h 

C3 

493.    It  should  be  noted  that  (§  461)  : 

The  common  denominator  of  the  three  roots  is  the  deter- 
minant composed  of  the  coefficients  of  x,  y,  and  z. 

The  numerator  of  the  value  of  x  may  be  obtained  from  the 
denominator  by  substituting  di^  dg,  and  dg  in  place  of  the  cor- 
responding coefficients  of  x. 

Similarly,  the  numerators  of  y  and  z  are  formed  by  substi- 
tuting respectively  di,  ^2?  ^^^  ^3  ^^  place  of  the  corresponding 
coefficients  of  y  or  z. 

Ex.    Solve  the  system 

x  —  2y-\'Sz  =  2, 

2x-Sz  =  3y 

x  +  y-^z  =  6. 


Hence  x 


1-2      3 
2      0-3 
1      1      1 

=  19. 

2-2      3 
3      0-3 
6      1      1 

»   y  = 

1 

2 

1 

2 

3 

6 

8 

-3 

1 

,    z  = 

1  -2 

2  0 
1      1 

2 

3 
6 

19 


19 


19 


Whence  x  =  3,  y  =  2,  2;  =  1. 


494.-  The  roots  of  a  system  of  four  or  more  simultaneous 
equations  can  be  obtained  in  exactly  the  same  manner. 


426 


ADVANCED  ALGEBRA 


495.    Elimination.     Eliminate  x  and  y  from  the  three  equa- 
tions ; 

aiaJ  +  &i2/  +  Ci  =  0,  (1) 


a2X  +  b^-\-C2  =  0, 
a^x  +  632/  +  Cg  =  0. 


(2) 
(3) 


Multiplying  the  equations  in  order  by  (7i,  C2,  and  Cg,  the  re- 
spective co-factors  of  Ci,  Cg,  and  Cg  in  the  determinant, 


D  = 


ttl 

&i 

<h 

a2 

h 

C2 

% 

h 

C3 

(4) 


and  adding, 

(aiOi+^aCi-f  a3(73)a;4-(&iCi-}-62C2+63Q2/4-Ci(7i-fC2(72-f  CgCa  =  0. 


Or,  considering  §  486, 


a. 

\ 

Ci 

a^ 

h 

C2 

ag 

h 

C3 

0. 


(5) 


This  equation  expresses  the  condition  which  must  be  satisfied 
by  the  coefficients  of  a^,  b^,  Cj,  etc.,  if  the  given  equations  are 
consistent. 


496.  The  left  member  of  equation  (5)  is  the  eliminant  of  the 
given  system  of  equations. 

497.  If  the  eliminant  is  a  quantity  which  cannot  be  equal 
to  zero,  the  equations  are  inconsistent. 

14     5  1 
Thus,  4  X  -f  6  =  0  and  3  a;  -|-  2  =  0  are  inconsistent,  since  2  r    ^' 

498.  Similarly,  we  may  eliminate  x,  y,  and  z  from  the  fol- 
lowing system : 

a^x  +  hyy  +  c^z  +  d^  =  0, 

OjfC  +  ^2^  4-  C22  +  c?2  =  0, 


DETERMINANTS 


427 


The  result  is 


a^x  +  642/  H-  C4Z  +  ^4  =  0. 

0^2       ^2  2  2 

«3       ^3       C3       (?3 
a4       64       C4       C?4 


=  0. 


Ex.  1.    Eliminate  x  smd  y  from  the  following  system : 

2x  +  y-\-a  =  0, 
x-2y  +  S  =  0, 
x-i-y-\-a=.0. 


The  eliminant 


Or,  expanding, 


2 

1 

a 

1 

-2 

3 

1 

1 

a 

=  0. 


-2a -3  =  0. 

Ex.  2.   Eliminate  a?  and  y  from  the  equations : 
3x-^2y-z  =  S, 
2x-2y-{-Sz  =  6, 

x-2z  =  -S. 

Transposing  the  right  members  and  applying  §  498,  we  have 

=  0. 


3  2  -z-S 
2  _2  Sz-6 
1  0    -2;^  +  3 


Expanding  and  simplifying,    0  —  2  =  0. 

Ex.  3.   Eliminate  x  from  the  two  equations 
a^ -{- xy -[■  y^  =  a% 
x  —  2y  =  —  2a. 


(1) 

(2) 


428 


ADVANCED  ALGEBRA 


Transposing  the  right  members  and  multiplying  (2)  by  x,  we  obtain 

the  three  equations 

x^  +  xy-\-y^~a-  =  0, 

x-2y  +  2a  =  0, 

x^-x(2y-2a)=  0. 


Eliminating, 


Expanding, 


1  y  y'^-  a2 

0       1  -2y  +  2a    =0, 

1     -2y-\-2a  0 


EXERCISE   156 
Solve  the  equations : 

x-\-2y-{-3z  =  6, 
[Sx-\-y-i-bz  =  9, 

x  +  y-z  =  17, 

x-\-z-y  =  13, 

I  -x-{-y-\-z  =  7, 

(x-\-2y-\-4.z  =  15, 

3.  x  +  y-\-z  =  9, 

[x  +  Sy-\-9z  =  23. 

tx-{-y  —  z  =  0, 

4.  \x  +  2y-\-z  =  Sj 
[3x  +  5y-z  =  10. 

(x  +  y-{-z  =  a  +  b-{-c, 
9.  \  ax -\- by  -^  cz  =  ab -\- ac -\-  5c, 

[  (b  —  c)x -\-  (c  —  a)y -}- (a  —  b)z  =  0, 

[  X  +  y  +  z  -{-  u  =  60j 


\x-2y  +  z  =  0, 
5.  -!  7x-{-6y-^7z  =  100, 
[3x-\-y-2z  =  0. 

i^x-\-y-\-z  =  26, 
7x-llz  =  0j 
9y-14a;  =  0. 

x  +  y-{-z  =  5j 

3x  —  5y  +  7z  =  7b, 

9  x  -  11  ;2  + 10  =  0. 

'  x  +  y  =  16j 
y+^  =  28, 

z-\-x  =  22. 


6. 


7. 


10.  { 


x  +  2y-{-3z-\-Au  =  100, 
x  +  3y-{-6z-\-10u  =  150, 
x  +  4:y-\-10z  +  20it  =  210. 


DETERMINANTS  429 

11.  Eliminate  x  from  the  following  equations: 

r523r  +  498a  =  0, 
l622a;4-497a  =  0. 

12.  Eliminate  x  and  y  from  the  three  equations : 

ax  +  by  — 2  =  0, 
x-y-l  =  0, 
I  a;  +  y  —  a  =  0. 

13.  Eliminate  x  and  y  from  the  following  equations,  and 
factor  the  result: 

( x-{- ay -\-a^  =  0, 
\x-\-by-{-b'  =  0, 
[  X -\- cy  -\- c^  =  0. 

14.  Eliminate  x,  y,  and  z  from  the  following  equations : 

ax+   2/  +  2!-fl  =  0, 
x  +  ay  =  0, 
y  +  z  =0, 

15.  Eliminate  x  and  y  from  the  following  three  equations: 

'a;  +  2/  +  2;  =  9, 
x-^2y  +  3z  =  14:, 
^x-\-Sy  +  7z  =  21. 

16.  Eliminate  y  and  «  from  the  following  equations : 

(a  +  b)x  -{-(a  —  b)z  =  2  be, 
(b  4-  c)y  +  (6  —  c)a;  =  2  ac, 
(c  -h  a)2;  -f  (c  —  a)y  =  2  a6. 


430 


ADVANCED  ALGEBRA 


17.   Eliminate  x,  y,  z  from  the  following  equations: 

3  a;  +  2/  -f-  2  =  20, 
a;  +  42/  +  3w  =  30, 
6aj  +  2  +  3t^  =  40, 
82/  +  3;2  +  5w  =  50. 


18.   Eliminate  x  from  the  following  equations : 

r2a;2 -22/2  =  2, 
l3aj  +  22/  =  2. 


Are  the  following  equations  consistent  ? 

a;2-j-3a.'-7  =  0, 
2a;-3  =  0. 


19. 


20. 


cc-l-Gy  —  5  =  0, 
2a;-32/-l  =  0, 
x  +  2/-2  =  0. 

3a;_l_82/-22  =  0, 
2a;-32/-2  =  0, 
a;  +  2/  ~  ^  =  ^• 


21. 


r2a^4-2a;  +  l  =  0, 
22     ^ 

[a;2  +  a;4-2  =  0. 

Hint.     Multiply  each  equation 
by  X,  and  eliminate  o;^,  x^,  and  x. 


CHAPTER  XXVIII 
THEORY  OF  EQUATIONS 

499.  General  form  of  an  equation  of  the  nth  degree.      E^ery 

equation  of  the  ?ith  degree  can  be  reduced  to  the  form : 

OqX''  +  ttioj"-!  4-  a^""-^  -\ h  a^.ix  +  a„  =  0.  (1) 

Dividing  every  term  by  ao,  and  substituting  p  for—,  we  obtain 
the  equation  of  the  nth  degree  in  its  simplest  form : 

X-  -hp^X^-^  +^235"-'  +   -   +Pn~l^  +Pn  =  0.  (2) 

Unless  stated  otherwise,  the  coefficients  pi,  Pi,  •••  i?„  are 
supposed  to  be  rational. 

500.  If  none  of  the  coefficients  p^,  P2,  •••  Pn  is  zero,  the  equa- 
tion is  complete,  otherwise  the  equation  is  incomplete. 

x^  —  3x^  +  405  —  7isa  complete  cubic  equation, 
ic^  —  2  X*  +  9  is  an  incomplete  equation. 

501.  In  the  present  chapter  we  shall  usually  represent  the 
rational  integral  function  of -{- piX''-'^ -\- p^p:?''^ -\-  -"  -\-Pn-\^+Pn 
by  j(x).     Thus  equation  (2)  may  be  written  f(x)  =  0. 

502.  A  commensurable  root  is  a  real  root  which  is  either 
integral  or  fractional  Roots  which  cannot  be  expressed  ex- 
actly by  integers  or  fractions  are  called  incommensurable  roots. 

SYNTHETIC  DIVISION 

503.  The  work  of  the  following  chapters  risquires  frequently 
the  division  of  a  rational  integral  function  by  a  binomial  of 

481 


432  ADVANCED  ALGEBRA 

the  form  x  —  a.     In  such  cases  the  work  of  division  can  be 
greatly  shortened  by  a  method  called  synthetic  division. 

,  Divide  Sx^-^x^-2Sx-{-9  by  x-4. 

Sx^-   4x2-23x  +  9|a;-4 


3  a;3  -  12  x^  |3x'^  +  8x+9 

+   8a;2_23a: 
+__8x^_-32^ 

+   9x-\-   9 

+   9  a; -36 

+  45 

This  work  may  be  abridged  by  omitting  the  literal  factors  (Appendix  I. 
Detached  Coefficients). 

1-4 


3  4-8  +  9 


3_  4-23  +  9 

3-12 

+    8-23 

+   8-32 

+    9+9 

+   9-36 

+  45 

The  second  terra  of  each  partial  product  may  be  omitted,  for  it  is 

merely  a  repetition  of  the  number  above,  and  by  changing  the  sign  of 

the  second  term  in  the  divisor,  every  subtraction  may  be  changed  to  an 

addition. 

3-   4- 23 +  911  +  4 


3  +  12  13  +  8  +  9 

+   8 

+  32 
+   9 

+ 


+  45 

Since  the  first  term  of  each  successive  remainder  is  equal  to  the  corre- 
sponding term  of  the  quotient,  we  may  omit  the  quotient.  Omitting  also 
the  first  term  of  the  divisor,  the  work  can  be  contracted  as  follows  : 

3  _   4  _  23     +9  I  +4 
+  12  +  32     +36 


Quotient  =  3+   8+9;  +  45  =  Remainder. 


THEORY  OF  EQUATIONS  433 

504.  To  divide,  therefore,  any  rational  integral  function /(a?) 
bj  x  —  a,  proceed  as  follows  ; 

Write  the  coefficients  of  x  in  a  horizontal  line,  representing  all 
missing  powers  of  x  by  zero  coefficients;  and  bring  doivn  the  first 
coefficient. 

Multiply  the  first  coefficient  by  a,  and  add  the  product  to  the 
second  coefficient. 

Multiply  the  resulting  sum  by  a,  and  add  the  product  to  the  next 
coefficient,  aiid  so  forth. 

Tlie  last  sum  is  the  remainder,  and  the  preceding  numbers  in 
oi'der  are  the  coefficients  of  the  quotient. 

Ex.1.    Divide2a^-45a^-9a;-7  by  a;-5. 

2+    0-45-    9   .-    7  I  +5 
4.104.50  +  25    +80 

2  +  10  +    5  +  16  ;  +  73 

The  quotient  is  2  a;^  +  10  x^  +  6  a;  +  16,  the  remainder  +  73. 

Ex.2.    Divide3a;^  +  3a^4-Tby  a;4-2. 

3  +  3  +  0,+   0+7  I  -2 
_  6  +  6  -  12    +24 

Quotient  =  3-3  +  6-12;  +31  =  Remainder. 


EXERCISE  167 

By  synthetic  division,  find  the  quotient  and  the  remainder  of 
each  of  the  following  divisions : 

1.  (a;^-2iB3  +  a^_2a;-hl)-i-(a;-2). 

2.  (2fl;*-3a:8-5a^-2fl?4-7)^(a;-3). 

3.  (a?*-3ar'-a;-f3)^(a;-l). 

4.  (2a^  +  3a^-2a^+5a;-7)--(a^H-3). 

5.  (3a^-2a^^-cc-6)-^(a;-4). 

6.  (3a:'-2aJ*4-2a5»-3aj2  4.2a;  +  2)-5-(a;+-2). 
2f 


434  ADVANCED  ALGEBRA 

7.  (2i^-3a^  +  2a5-l)--'(a;-4). 

8.  (x'-6a^-{-4:x'-2x-7)-h(x-5).' 

9.  (a;^4-6a^  +  4a^  +  2a;  +  7)-^(a;  +  5). 

10.  (x^-12)^(x-l). 

11.  (x^  +  x^-}-x^-\-x^-\-x^-\-x  +  l)-i-(x-l), 

12.  (x«+3a^-9)--(a;  +  2). 

13.  (6x'-5a^-{-2x^-9x-7)^(x-3). 

14.  (aj3-2aa^  +  2a2a;  +  a3)--(a;-a). 

APPLICATION  OF   THE   FACTOR  THEOREM 

505.  If  f{x)  =  aQX^ -\-aiX''~'^ -\- '" -{-a^,  and  a  is  a  root  of 
the  equation  f(x)  =  0,  then  f(x)  is  divisible  by  x  —  a.  (Factor 
Theorem.) 

The  proof  which  was  given  in  Chapter  XVI  may  be  briefly- 
restated  as  follows : 

Divide  f(x)  by  (x  —  a)  until  the  remainder  B  no  longer 
involves  x.     Denoting  the  quotient  by  Q(x),  we  have 

{x-a)Q{x)-\-Ii=:f(x).  (1) 

Substituting  a  for  a;,  E  =  0. 
I.e.  f(x)  is  divisible  by  a;  —  a. 

E.g.  if  f(x)  =  X*  ~  6x^  +  x^  +  2x  +  1,  then  /(I)  =  0.  Hence  x-1 
is  a  factor  of  /(x) . 

506.  Conversely,  if  a  rational  integral  function  f(x)  is  divisible 
by  x  —  a,  then  a  is  a  root  of  the  equation  f(x)  =  0.  (1) 

For  if  Q(x)  is  the  quotient  obtained  by  dividing  f{x)  by 
x  —  a,  equation  (1)  becomes 

(x-a)Q(x)=0, 

This  equation  is  obviously  satisfied  by  a;  =  a. 


•      THEORY  OF  EQUATIONS  435 

Ex.    1.  Prove  that  5  is  a  root  of  the  equation 

Dividing  by  a;  -  5,  l  _  3  -  18  +  40  |_5 

+  6  4-  10  -  40 

1  +  2-    8;      0 

Since  there  is  no  remainder,  5  is  a  root  of  the  given  equation. 

507.  If  f(x)  is  divided  by  x  —  a,  the  remainder  of  the  division 
is  equal  tof{a).     (Remainder  Theorem,  Chapter  XVI.) 

Using  the  notation  of  §  505,  we  have 

(x-a)Q(x)-\-Ii=f(x). 
Substituting  x  =  a,  R  =f(a). 

Ex.2.    lff(x)  =  x*-2a^-9x'  +  2,^jidf(^y 

Dividing  /(x)  by  x  —  4, 

1-2-9  +  0  +  2  |_4 

+  4  +  8-4-16 
1  +  2-1-4;  -14 

Hence  /(4)  =  -  14. 

508.  The  preceding  method  of  substitution  may  also  be  dem 
onstrated  as  follows : 

x*-2x8-9x2  +  0.x+    2|_4 
+  4x«  +  8x2-    4x     -16 

X*  +  2  x8  -     x2  -    4  X ;  -  14 
I.e.  if  X  =  4, 

X*  =     4  X*,  which  added  to  the  next  term  gives  +  2  x*. 

2  X*  =     8  x2,  which  added  to  the  next  term  gives  —  x^. 

—  x2  =  —  4  X,  which  added  to  the  next  term  gives  —  4  x. 

—  4x  =  — 16,     which  added  to  the  next  term  gives  —  14. 

Therefore  /(4)  =  -  14. 


436  ADVANCED  ALGEBRA 

EXERCISB   168 

1.  Prove  that  a;  -1  is  a  factor  of  66  ar*  -  66  a;*  +  73  a^  -  70  a^ 
+  4  a;  — 7. 

2.  Prove  that  a;  - 1  is  a  factor  of  637  a;"^  -  638  x^^  + 1. 

3.  Prove  that  a;  -  a  is  a  factor  of  6  a;"-8  a;V-4  a^ai2_^6  a>\ 

4.  Prove  that  a  +  2  6  is  a  factor  of  a^  — 16  h\ 

Prove  the  following  statements  by  synthetic  division: 

5.  -2  is  a  root  of  a^-2a;*-3a^  +  3a^-l2a;4-4  =  0. 

6.  3  is  a  root  of  a^  +  a^  -  17  a^  + 17  a;  -  6  =  0. 

7.  -2  is  a  root  of  a«  +  4ar^  + 7a;^  +  10ar'  +  13a7*  +  16a;  +  12 
=  0. 

8.  -5  is  aroot  of  a;^  + 7a:^  + 7a;2- lla;  + 20  =  0. 

9.  3  is  a  root  of  3a;*- 6a;5 -2aj2- 20aj-3  =  0. 

10.  -5  is  a  root  of  2  a;* -f- 13  ar*  +  20  a;^  +  31  a;  +  30  =  0. 

11.  -6  is  a  root  of  4ar'+19a;*-27  x^H-20a;^+13a;-|- 6  =  0. 

12.  3  is  a  root  of  5  a;^-  18  a;*  +  H  a;^  -  10  aj^-f- 13  a;  -  3  =  0. 

13.  -5  is  a  root  of  6 a;»  + 15 a;* - 55 a;^ -f  64 a;* - 136 a;  +  220 

=0. 

14.  If /(a;)=3a:»  +  2a.-«-3a;-5,  find/(2). 

15.  If /(a;)=2a^-3a^-f  4a;~-7,  find^-2). 

16.  If /(a;)  =  4a;*-5ar^  +  2ar^-7a;-9,  find/(3). 

17.  If /(«')  =  6 a;«- 2 a.-^- 7,  find /(-I). 

18.  If /(a;)  =  4a;*-3a:2_7^^2,  find/(6). 

19.  If /(a;)  =  6a:^-18a;*  +  3a;^-10a;-2,  find/(3).    . 

20.  If  fix)  =  a;*  -  5  aa;^  +  2  aV  - 17  a?x  -  41  a^  find  j{3  (^^ 


THEORY  OF  EQUATIONS  437 

NUMBER  OF   ROOTS 

509.  We  shall  assume -the  fundamental  theorem,  that  every 
rational,  integral  equation  has  at  least  one  root.  The  proof 
of  this  proposition  is  beyond  the  scope  of  this  book.* 

510.  Every  rational,  integral  equation  of  the  nth  degree  has  n 
roots. 

If  f(x)  =  ic'*  -hpicc"-^  +p2^"-2  -h  ...  -i-Pn-iX  +p„,  and  r^  is  a  root 
of  the  equation  f(x)  =  0,  then  f(x)  is  divisible  by  a;  —  ri  (§  505). 
Denoting /(ic)  -i-(x—  n)  by  fi(x),  we  have 

f(x)==(x-r,)fi(x). 

But  fi(x)  is  a  rational,  integral  function  of  the  (n  —  1)  degree, 
hence  it  has  a  root.  Let  this  root  be  r^,  then  we  obtain  in  a 
similar  manner  as  before  fi{x)  =  (x^r^f2{x),  where /2(aj)  is  of 
(n  —  2)  degree. 

Therefore  f{x)  =  {x  —  r-^  {x  —  r^f2,{x) . 

Continuing  this  process,  f{x)  can  be  resolved  in  n  factors, 
viz.  x  —  ?'i,  x  —  r2"'X—rn' 

Or  f(x)  =  {x-r^{x^r2)(x-r^  ...  («-r„). 

Hence  the  equation  f(x)  has  n  roots,  for  f(x)  vanishes  when 
X  is  equal  to  any  of  the  values  ri,  rg,  rg  ♦•.  r„. 

Note.     If  F(x)  =  aoX"  +  oiX'-^  '-an,  it  can  easily  be  shown  that 
F{x)=  ao{x-r{)  (x  —  r2)(x  -  rs)  ...  (x  -  r„). 

511.  Multiple  roots.  The  n  roots  of  an  equation  of  the  nth 
degree  are  not  necessarily  all  different.  E.g.  the  equation 
(a;  — 4)(a;  — 4)(a;  — 3)  =  0  has  the  roots  4,  4,  3.  Every  root 
occurring  more  than  once  is  called  a  multiple  root ;  thus  4  is  a 
double  root. 

In  such  cases,  however,  the  roots  are  counted  as  if  they  were 
all  different. 

«  See  Bumside  and  Fauton,  Theory  of  Equations,  p.  259. 


438  ADVANCED  ALGEBRA 

512.  Depression  of  an  equation.  If  r^  is  a  root  of  the  equa- 
tion f(x)  —  0,  we  can,  by  dividing  f{x)  by  a;  —  r^,  reduce  or 
depress  the  equation  to  one  of  the  next  lower  degree,  which 
contains  all  the  remaining  roots. 

E.  g.  the  equation  x^  —  5  x^  —  9  x  +  45  =  0  has  one  root  equal  to  3. 
Dividing  by  x  —  3,  we  obtain  the  depressed  equation  x^  —  2  x  —  15  =  0, 
whose  roots  are  5  and  —  3. 

Hence  5,  —  3,  and  3  are  the  roots  of  the  given  equation. 

513.  Solution  by  trial.  If  all  roots  but  two  of  an  equation 
are  integers,  it  is  often  possible  to  solve  the  equation  by  trial. 

Ex.    Solve  a;^-f4aj^-3a:3_32ari_54a;_36  =  0. 

If  there  are  integral  roots,  they  must  be  factors  of  36 ;  i.e.  ±1,  ±  2, 
±  3,  etc.  (§  625). 

Substituting  + 1,  1+4-3-32-54-36=5^0. 

Substituting  -1,      -1  +  4  +  3- 32 +  54- 36  ^fcO. 

Hence  +  1  and  —  1  are  not  roots. 
Dividing  by  x  —  2, 


1 

+  4 

-3 

-32 

-54 

-36 

\1 

+  2 

+  12 

+  18 

-28 

-164 

1 

+  6+9 

-14 

-82; 

-200 

Hence  2  is  not 

a  root. 

Dividing  by  x 

+  2, 

1 

+  4 

-3  - 

-32  - 

-54  - 

-36  1- 

2 

-2 

-4 

14 

4-36  +36 

1+2-7-18-18;       0 

Therefore,  —  2  is  a  root,  and  the  depressed  equation  is 

a:4  +  2x8-7x2-18x-18  =  0. 

As  -  2  may  be  a  double  root,  we  divide  again  by  x  +  2,  but  obtain  a 
remainder. 

Dividing  by  x  —  3, 

^    ^       ■     '     1+2-7     -18  -18  [3 

+  3  +15  +24  +18 
1   +5  +8     +6;         0 

That  is,  x=3  is  another  root,  and  the  next  quotient  is  x^  +  S  x2+8  x+6. 


THEORY  OF  EQUATIONS  489 


Dividing  by  x  + 3,  ^  ^^  _^  g  _j.6|__j 

-3  -6  -6 


1  +2  +2;    0 


Therefore,  a;  =  —  3  is  a  third  root,  and  the  last  quotient  is  x'^ -\- 2  x -\- 2, 
which  cannot  be  factored. 

Solving  x2+2x  +  2  =  0by  formula,  we  obtain  a;  =  — 1±V— 1. 
Hence  the  five  required  roots  are  —  2,  +3,  —  3,  —  1  +  i,  —  1  —  i. 

514.   Formation  of  equations.     If  all  roots  of  an  equation  are 
given,  the  equation  can  be  formed  by  inspection. 

Thus,  the  equation  whose  roots  are  —  2,  4-2,  and  +  3  is 

(x  +  2)(x-2)(a;-3)  =  0. 

Or  a:»- 3x^-4x4- 12  =  0. 

EXERCISE  159 

Solve  the  following  equations : 

1.  0^  +  6x^  +  10  x-^S  =  0,  one  root  being  —  4. 

2.  a^  +  7  a^  +  7  a;  — 15  =  0,  one  root  being  —  3. 

3.  a^  —  5  a^  —  9  a;  H-  45  =  0,  one  root  being  5. 

4.  a:^  4- a^  —  a;  —  10  =  0,  one  root  being  2. 

5.  2a^  +  7a^  +  2;c  —  3  =  0,  one  root  being  —  1. 

6.  3  a:^- 16  a^  + 23  a;- 6  =  0,  one  root  being  2. 

7.  2a^  +  3a^-29a;  +  30  =  0,  one  root  being  - 5. 

8.  a;*  —  2  a;2  —  3  a;  *—  2  =  0,  two  roots  being  —  1,  2. 

9.  3  a;*- 16  a^+ 14  x^-{-  24  a;  -  9  =  0,  two  roots  being  -1,3 

Solve  by  trial : 

10.  a^-7ar^  +  16a;-12  =  0. 

11.  ar'-12a^  +  47a;-60  =  0. 

12.  a;»-10a:2_,_3i3._30::=0. 


440  ADVANCED  ALGEBRA 

13.  a.'3-8a;2  4-19a;-12  =  0. 

14.  a'3-6a^  +  lla;-6  =  0. 

15.  a^  +  9aj2  +  27a;  +  27  =  0. 

16.  a;^-10a:3  +  35ic2_50aj4-24=:0. 
IT.  2a;*-3a^-12a^  +  7ic  +  6  =  0. 

Form  the  equations  whose  roots  are  ; 

18.  1,-2,3,-4.  22.    ±VE,  ±Q. 

19.  1,  2,  3,  0.  23.    3,  2,  1  -f- V2,  1  -  V2. 

20.  2H-V5,  2-V5,  3.  24.    -1,  +  1,  1 +^,  1 -i 

21.  ±2,  ±V5.  25.    ±V2,  2±V2. 

26.  How  many  roots  has  the  equation  x^  —  1? 

27.  How  many  roots  has  the  equation  VaJ  =  1? 

RELATIONS  BETWEEN  THE  ROOTS  AND  THE 
COEFFICIENTS 

515.   The  equation  whose  roots  are  rj,  rg,  rg,  is 

(a;  —  rj)  (aj  —  rg)  (a;  —  ?'8)  =  0. 

Or,  expanding, 

^—{n  +  n  +  n)  ^^  +  (n^2  +  nr^  +  rzVs)  x  -  r^Tzr^  =  0. 

Comparing  this  result  with  the  general  cubic  equation  in  its 
simplest  form,  viz. : 

a^-{-PiX^-\-p^-\-Ps  =  0,  we  obtain 

^i4-r2  +  r8  =  -i)i. 

rir2  +  rin-hr2n=     Pi- 

nnr^^-Pz. 


THEORY  OF  EQUATIONS  441 

516.    In  a  similar  manner  we  obtain  for  n  roots, 
(aj- n) (a; - r2)(a?- rg)  •••  (a;  - r„)  =  0. 
Or,  expanding, 
a?"  -  (n  4-  ^2  +  ••'  r„)af-i  +  (rirg  +  rir^  +  r^r^  +  nn  +  -O^"^ 

-  C''i^2^'3  +  r^r^n  +  n*'3»*4  +  •  •  •)  a?**"^  H (  -  '^Tn^n  •  •  •  r«  —  0. 

The  general  equation  for  the  nth  degree  in  its  simplest  form  is 

«»  +p^3r~''  +p^-^  +  •  •  •  +  Pn-i^  +P„  =  0. 
Comparing  the  coefficients,  we  have 

n  +  n  +  ^3--+^„  =  -Pi. 


517.   I.e.  in  an  equation  in  its  simplest  form : 

1.  Hie  sum  of  the  roots  is  equal  to  the  coefficient  of  the  second 
term  with  its  sign  changed. 

2.  TJie  Slim  of  the  products  of  the  roots  taken  two  at  a  time  is 
equal  to  the  coefficient  of  the  third  term. 

3.  The  sum  of  the  products  of  the  roots  taken  three  at  a  time  is 
equal  to  the  coefficient  of  the  fourth  term  with  its  sign  changed,  etc. 

4.  Tlie  product  of  all  roots  is  equal  to  the  last  term,  and  is 
positive  or  negative  according  as  the  degree  of  the  equation  is  even 
or  odd. 

E.g.  itx^  +  2x-h  6  =  0,  the  three  roots  vi,  r^,  r»,  satisfy  the  following 

equations : 

ri-h  r2  +  r^  =  0. 

fira  +  rirs  +  r^ra  =  2. 
riViTs  =  —  5. 


442  ADVANCED  ALGEBRA 

518.  The  following  special  cases  should  be  noted : 

1.  If  the  second  term  is  wanting,  the  sum  of  the  roots  is  zero. 

2.  If  the  absolute  term  is  wanting,  at  least  one  root  is  zero. 

519.  The  relations  between  the  roots  and  the  coefficients 
cannot  be  used  to  solve  the  equation,  as  the  work  always  leads 
to  the  original  equation. 

E.g.  let  «»  -  3  aj2  4.  2  x  +  5  =  0. 

Then  ri  +  r2  +  rg  =  3. 

rir2  +  riVs  +  VoVs  —  2. 

nr^n  =  —  5. 

Eliminating  r^  and  7*3,    j'l^  —  3  ri^  +  2  ri  +  5  =  0. 

520.  If,  however,  certain  relations  exist  between  the  roots,  the 
equation  can  frequently  be  solved  by  means  of  the  preceding 
propositions. 

Ex.  1.  Solve  4:0^-12  ay^ -^11  x-3  =  0j  if  the  roots  are  in 
arithmetical  progression. 

Let  r  —  s,  r,  r  -\-  s,  be  the  three  roots. 
Reducing  the  equation  to  its  simplest  form, 

4        4 

Therefore  3r  =  3.         .  (1) 

3  r2  -  s2  =  -1^.  (2) 

r(r2-s2)  =  f.  ,         (3) 

Solving  (1)  and  (2),  r  =  l,   s  =  \. 

Since  these  values  also  satisfy  equation  (3) ,  the  required  roots  are  ^, 
1,  f. 

Ex.  2.  Determine  n  so  that  one  root ofa^  —  7a^4-wa;  —  8  =  0 
is  the  double  of  another  root,  and  solve  the  equation. 


THEORY  OF  EQUATIONS  443 

Let  ri,  r2,  and  2  ri  be  the  three  roots. 

Then  3  n  +  r2  =  7.  (1) 

3  rin  +  2  ri2  =  n.  (2) 

nVa  =  4.  (3) 

From  (1)  ra  =  7  -  3  n. 

Substituting  in  (3)  and  simplifying, 

3  ri8  -  7  ri2  +  4  =  0. 
Solving  by  trial,  n  =  1,  2,  or  —  |. 

Hence  r2  =  4,  1,  or  9 ; 

2ri  =  2,  4,  or  -f ; 
and  w  =  14,  14,  or  —  17^. 

Ex.  3.  In  the  equation  ic^  +  aa^  +  6a;  —  c  =  0,  find  the  condi- 
tion that  the  sum  of  two  roots  is  zero. 

Let  ri,  r2,  and  —  vi  be  the  three  roots. 

Then  r2  =  —  a,  ri^  =  —  &,  n^ra  =  c. 

Eliminating  ?'i,  r2,  ab  =  c. 

521.  A  function  is  symmetrical  with  respect  to  two  letters  if 
an  interchange  of  the  two  letters  does  not  change  the  function. 

a^  -\- b^,  afi  +  S  xy  +  y^,  are  symmetrical  functions. 

522.  A  function  is  symmetrical  with  respect  to  three  or 
more  letters,  if  an  interchange  of  any  two  of  these  letters 
does  not  change  the  function. 

abc,  a"^  +  b^  -{•  c'^,  a  +  b  -^  c  +  a%'^c^,  are  symmetrical. 

523.  Symmetrical  functions  of  the  roots  of  an  equation  can 
be  found  without  solving  the  equation. 


444  ADVANCED  ALGEBRA 

Ex.  4.  If  7\,  Vz,  rs,  are  the  roots  of  aj^  +  Sa;'  —  2a;-f-5=s0, 
find  ri^  +  rs^ +  9-32. 

ri^  +  r<^  +  n^  =  (ri  -{- r2  +  ny  -  2(rira  +  nrs  +  rzrs) 
=  (_3)2-2(-2). 
Or,  »'i2  +  rg^  +  rs^  =  13. 

EXERCISE  160 

Solve  the  following  equations : 

1.  a^  —  2a^  —  9ic  +  18  =  0,  the  sum  of  two  roots  being  zero. 

2.  aj^  —  5flj^H-3fl7  +  9  =  0,  two  roots  being  equal. 

3.  4a^  —  32a;^  —  a;  +  8  =  0,  the  sum  of  two  roots  being  zero. 

4.  aj»  —  3  a^  — 13  a;  + 15  =  0,  the  roots  being  in  A.P. 

5.  a^  +  a?^  — 10  a;  +  8  =  0,  one  root  being  twice  another. 

6.  a^  —  2a^— 5a;  +  6  =  0,  one  root  being  three  times  another. 

7.  a?^  —  8a;^  +  19a;  — 12  =  0,  one  root  being  equal  to  the  sum 
of  the  other  two. 

8.  a^  —  3a;2  —  6a;  +  8  =  0,  one  root  being  one-half  the  sum 
of  the  other  two. 

9.  a;^  -  9  a;2  +  26  X  -  24  =  0,  the  roots  being  in  A.P. 

10.  ar^  +  7  »2  + 14  a;  4-  8  =  0,  the  roots  being  in  G.P. 

11.  a^  - 14  a^  +  56  a;  -  64  =  0,  the  roots  being  in  G.P. 

12.  aj^  +  5a^  —  4a;  —  20  =  0,  the  sum  of  two  roots  being  zero. 

13.  Determine  n  so  that  one  root  of  the  equation  a^  — 3a;^ 
— 10  a;  +  71  =  0  is  the  double  of  another,  and  solve  the  equation. 

14.  Determine  n  so  that  the  sum  of  two  roots  of  the  equa- 
tion a^  —  5a^  —  4a;-fn  =  0,  is  equal  to  zero. 

15.  The  equation  a^  — aa;^H- &a;  — c  =  0  has  two  roots  whose 
sum  is  zero.  Find  (a)  the  third  root,  (b)  the  other  two  roots, 
(c)  a  relation  between  the  three  coefficients. 


THEORY  OF  EQUATIONS  446 

If  r,,  Tj,  7*3,  are  the  roots  of  the  equation  (x^ -\- ax^  +  bx  +  c  =t  0, 
find  without  solving : 

16.  ri  +  rj  +  rg-Sj-irarg. 

17.  (ri-^r2-\-rQy-4:(nr2-i-rirs  +  r2n). 

18.  7-12+7-22  + 7-32. 

19.  (n  +  r^y  +  (r^  4-  r,y  +  (7-3  +  r,y, 

20.  In  the  equation  ic^  —  2a^  —  9a;  +  l  =  0,  find  the  sum  of 
the  squares  of  the  roots. 

21,  If  one  root  of  the  equation  fl^  +  aiB'  +  6aj  +  c  =  0  is  the 
reciprocal  of  another  one,  find  the  third  root. 

INCOMMENSURABLE   AND   IMAGINARY  ROOTS 

524.  If  all  coefficients  of  a  rational,  integral  equation  are 
integers,  and  the  first  coefficient  is  unity,  the  roots  cannot  he 
fractional. 

r 
Let  -  be  a  root  of  the  equation 

X-  +  p^x--"^  +  p^^--"  +  . . .  +  p„  =  0,  (1) 

and  suppose  that  r  and  s  have  no  common  factor. 
Substituting, 

i)"+^<ir+^<5r+-+^"=«- 

Multiplying  by  s~-^  and  transposing, 

~  =1  -  ;)i7'"-^  -  p^V'-h  -  i)8r"- V i)n«""^ 

s 

which  is  impossible,  as  the  left  member  is  a  fraction  in  its 
lowest  terms,  and  the  right  member  is  an  integer. 
Hence  equation  (1)  cannot  have  fractional  roots. 


446  ADVANCED  ALGEBRA 

525.  If  the  coefficients  of  a  rational,  integral  equation  are 
integers,  all  integral  roots  are  factors  of  the  absolute  term. 

Suppose  r  is  an  integral  root  of  the  equation 

aoX""  +  a^x''-'^  +  a<pc''-^  -\ f-  a„_-^x  +  a„  =  0. 

Substituting,  transposing,  and  dividing  by  r, 

aor*»-i  +  tti^""^  +  «2^*'*~^  H h  «n-i  =  —  — * 

r 

Since  the  left  number  is  an  integer,  the  right  member  must 
also  be  an  integer ;  i.e.  r  must  be  a  factor  of  a„. 

526.  If  a  complex  number  is  a  root  of  an  equation  with  real 
coefficients,  f{x)  =  0,  its  conjugate  is  also  a  root. 

Let  a  +  hi  be  a  root,  and  Q  —f{x)  -=-  [a:  —  (a  +  hi)']. 

Then  lx-{a-^hi)']Q=f{x).  (1) 

As  Q  involves  ?',  we  would  obtain  a  different  expression,  Q', 
if  we  would  substitute  —  i  in  place  of  i. 

Since  the  product  of  x—  {a-{- hi)  and  Q  (i.e.  f(x))  is  real,  it 
contains  only  even  powers  of  i. 

Hence,  if  we  substitute  —  i  in  place  of  i  in  the  left  member 
of  (1),  the  product  n^ust  be  the  same  as  before,  as  the  even 
powers  of  i  and  —  i  are  identical. 

Hence  [x—(a  —  hi)]  Q'  =f(x). 

I.e.  a  —  hi  is  a  root  of  f(x)  =  0. 

527.  To  every  pair  of  imaginary  roots  there  corresponds  the 
quadratic  factor  (x  —  a)^  +  h^,  which  is  real  and  positive  for  any 
real  value  of  x. 

Hence  any  rational,  integral  function  of  x  can  be  resolved 
into  real  factors,  which  are  either  linear  or  quadratic  in  x. 


THEORY  OF  EQUATIONS  447 

528.    If  a-\-  -y/b  is  a  root  of  an  equation  whose  coefficients  are 
rational,  a  —  V6  is  also  a  root. 

The  proof  is  similar  to  that  of  §  526. 
Ex.  1.    Solve  the  equation 

one  root  being  2  —  V  — 3. 

Since  2  —  V—  3  is  a  root,  2  +  V—  3  is  also  a  root. 
The  quadratic  factor  corresponding  to  these  roots  is 

(a:  -  2  +  >/^)(x -  2  -  V^),  or  x'^-ix  +  7. 

Removing  this  factor  by  division,  we  obtain  the  depressed  equation 
x^  —  a;  —  6  =  0,  whose  roots  are  —  2  and  3. 

Hence  the  four  roots  are  2  —  V—  3,  2  +  V—  3,  —  2,  3. 

Ex.  2.    Form  an  equation  with  rational  coefficients,  one  of 
whose  roots  is  V2  +  V3. 

Since  V2  +  V3  is  a  root,  V2  —  VS  must  be  a  root. 
Similarly,  -  \/2  +  V3  and  —  V2  —  V3  are  roots. 
Hence  the  required  equation  is 

(a;  _  V2  _  V3)(a;  _  V2  +  V3) (x  +  \/2  -  y/S)(x  +  >/2  +  V3)  =  0. 
Or  ic*  -  10  x2  +  1  =  0. 

EXERCISE  161 

Solve  the  following  equations,  having  given  the  indicated 
root: 

1.  a^_2a;-4  =  0;  -l+V^^l. 

2.  af'-h48a;  +  504  =  0;  3  +  5V^. 

3.  a^-21x-3U  =  0;  _4  +  3V^^. 

4.  a^-9a;-|-28  =  0;  2-\-V^~3. 


448  ADVANCED  ALGEBRA 

6.  a^-7a;2-12a!4-18  =  0;  _2  +  V^^. 

7.  a;^-10.T^  +  33a;2_4g^_^20  =  0;  3  +  V5. 
•      8.    a;4-8a^  +  14.T2^4aj_8  =  0;                         14-V3. 

9.    x(a^4-l)(a^  +  2)(.T  +  3)=24;  _|  +  iV^Zi5. 

10.  If  -2  +  V^^  is  a  root  of  ic* -f2a^- 16a;  +  77  =  0, 
resolve  the  left  member  into  real  factore. 

11.  If  2  +  i  is  a  root  of  a^  —  3  aj^  +  a;  +  5  =  0,  resolve  the  left 
member  into  reaj  factors. 

12.  Can  7  be  a  root  of  the  equation  a^-f2a:2H-5a;  +  12  =  0? 
Can  ^  be  a  root  of  the  same  equation  ? 

13.  Form  an  equation  with  rational  coefficients,  one  of  whose 
roots  is  ■\/2  4-  i. 

14.  Form  an  equation  with  rational  coefficients,  one  of  whose 
roots  is  V3  —  2  i. 

TRANSFORMATION  OF  EQUATIONS 

529.  The  solution  of  an  equation  is  sometimes  simplified  by 
transforming  it  into  another  one  whose  roots  bear  a  certain 
relation  to  the  original  one. 

530.  The  method  used  for  most  transformations  consists  in 
expressing  the  relation  between  x  and  the  new  variable  y  in 
the  form  of  an  equation.  Find  x  in  terms  of  3/,  and  substitute 
this  value  in  the  given  equation. 

Ex.«l.  Find  an  equation  whose  roots  are  the  reciprocals  of 
the  roots  of  the  equation  2ic^  +  3ic^  +  4a^  +  5a;  +  6  =  0. 

Make  w  =  -,  then  a;  =  — 

a;  y 

Substituting,  l  +  l  +  i  +  §4-6  =  0. 

y4       y^       y^       y 

Multiplying  hy  f,   6f -\- 5y^  r\- iy^ '\- Sy +  2  =  0, 


THEORY  OF  EQUATIONS  449 

Ex.  2.   Find  an  equation  whose  roots  are  the  cubes  of  the 
roots  of  the  equation  f(x)  =  0. 

Make  y  =  a;^,  then  x  =\/y,  therefore  f{y/y)  is  the  required  equation. 

Ex.  3.   Find  an  equation  the  cubes  of  whose  roots  are  seven ' 
times  the  squares  of  the  roots  of  the  equation  f(x)  =  0. 


Let  2/3  =  7 cc2,  then  x  =  J^,  therefore  fi\-\  =  0. 


531.  Tramform  the  equation 

x""  +pia;«-^  -^-p^""-^  H \-Pn-i^  -\-Pn  =  0 

into  another  whose  roots  are  those  of  the  given  equation  with 
their  signs  chariged. 

Let  y  =  —  x,  then  x  =  —  y. 

Substituting, 

(-  yy  +Pi{-  yT-\+P2{-  yy-'  +  -  +p.-i(-  y)  +Pn  =  0. 

Simplifying,  and  dividing  by  —  1  if  ri  should  be  odd, 

E.g.  to  change  the  sign^  of  the  roots  of  the  equation 
x^  -4x*  +  3x3  +  2x  -  5  =  0, 
change  the  signs  of  all  even  powers, 
i.e.  a:* +  4x4  +  3x3  +  2x4-5  =  0. 

532.  To  transform  the  equation 

af  +pia;"-i  ^-p^x^-^  +  •••  +P,.  =  0 

into  another  one  whose  roots  are  equal  to  m  times  the  roots  of 
the  given  equation. 

Let  y  =  mXj  then  x  =  —- 

2o 


450  '  ADVANCED  ALGEBBA 

Substituting, 

Multiplying  by  m", 

2/"  + 1^12/"  ~^^  4-  PiU'^'V  -\ 4-  jp„m'"  =  0. 

I.e.  the  required  equation  is  obtained  by  multiplying  the  second 
term  by  m,  the  third  by  rn\  etc. 

Thus,  the  equation  whose  roots  are  ten  times  the  roots  of 

a;3-4a;2  +  2x-  7  =  0, 

is  ic3_40a;2-|.200a;-7000  =  0. 

533.    The  principal  application  of  the  preceding  transforma- 
tion consists  in  clearing  equations  of  fractions. 

Ex.  1.   Transform  the  equation 
into  another  one  with  integral  coefficients. 

V 
Substituting  x  =  — , 

Let  w  =  6,  then  y^  -  Iby"^  -\- 1  y  -  \b  =  0, 

Ex.  2.   Transform  the  equation 

into  another  one  which  has  integral  coefficients  and  unity  for 
the  coefficient  of  o?. 

Dividing  by  16,  x^  -  \x'^  +  \x  +  t\  =  0. 

Let  x  =  ^,  y^-\y'^'>n-\-\ym^-\-  ^^  m^  -  0. 

in 

Let  wi  =  4,  i.  e.  x  =  ^« 
4 
Then  y8  -  y2  +  2  y  +  12  =  0. 


THEORY  OF  EQUATIONS  451 

Ex.  3.    Solve  the  equation 

a^-ix'  +  4:X-i  =  0,  (1) 

Letic  =  ^,  y^  -  7  y^+Wy  -12  =  0.  (2) 

If  there  are  any  rational  roots,  they  must  be  factors  of  12,  i.e.  ±  1, 
±2,  ±3,  etc. 

As  ±  1  evidently  does  not  satisfy  (2),  try  y  =  2. 


1  _  7  +  16  -  12  |2 

2-10  +  12 

1-5+    6        0 

Hence 

(y_2)(2,2_5y^.6)  =  0. 

Or 

(y-2)(y-2)(y-3)  =  0. 

I.e. 

y  =  2,  2,  or  3. 

Hence 

x  =  l,  1,  or  f. 

, 

EXERCISE  162 

1.  Transform  the  equation  oif  —  7x^  +2 a^+2  a^  —  Sx  —7=  0 
into  another  one  whose  roots  are  those  of  the  given  equation 
with  signs  changed. 

2.  Transform  the  equation  £c^— 7a^+2ic— 9=0  into  another 
one  whose  roots  are  five  times  the  roots  of  the  original  equation. 

3.  Transform  x^-2x^-{-3x^-4:X^-\-5x^-6x  +  7  =  0  so 
that  its  roots  equal  ten  times  the  original  roots. 

Transform  the  following  equations  so  that  the  first  coeffi- 
cient is  unity,  and  all  coefficients  are  integers : 

4.  a^-|a^4-|a;-i  =  0.  6.    a^-^a^-j^x  +  ^^O. 

5.  a^-^a^^7^x-^  =  0.  7.    x" -^a^-ix-^^O. 

8.  x*-3x^-ix''  +  ix  +  ^  =  0. 

9.  2x^-4:X^-3x^-f2x-i=:0. 
10.    3a^-6a^  +  4a;-4  =  0. 


452  ADVANCED  ALGEBRA 

Solve  by  trial : 

11.  ar^-3a^  +  -VaJ-f  =  0.  14.  ^:i? -Ux" ^  x  +  2  =  ^. 

12.  a33  +  |.T2  +  -^aj-2  =  0.  15.  8a;^- 12a;2-4a;+6=0. 

13.  x3  +  tar^-%^-a?+|  =  0.  16.  a^-2«2  +  ia-^=  0. 

DESCARTES'   RULE   OF  SIGNS 

534.  In  a  series  of  algebraic  numberSj  a  permanence  is  the 
succession  of  two  like  terms ;  a  variation  is  the  succession  of 
two  unlike  terms. 

Thus,    +H contains  two  permanences  and   one  variation,  and 

-\ 1 1 h  contains  one  permanence  and  six  variations. 

535.  Descartes'  Rule  of  Signs.  The  number  of  positive  roots  in 
an  equation  cannot  exceed  the  number  of  variations,  and  the  mim- 
ber  of  negative  roots  cannot  exceed  the  number  of  permanences. 

To  prove  the  theorem  for  positive  roots,  it  is  only  necessary 
to  show  that  for  the  introduction  of  each  positive  root,  at  least 
one  variation  is  added. 

Suppose  -the  signs  of  a  complete  equation  to  be 

+  +  -4- +  -. 

To  introduce  a  positive  root  a,  we  have  to  multiply  by  a;  —  a. 
Writing  only  the  signs,  we  have 

+  +  -  + +  - 

--  +  -4-  +  +  -  + 

4-±-  +  -±±  +  -  + 

In  the  result  the  ambiguous  sign  ±  is  used  wherever  the 
sign  is  doubtful. 

Comparing  the  signs  of  the  original  expression  with  the  signs 
of  the  product,  we  observe  the  following  facts,  which  can  easily 
be  proved : 


THEORY  OF  EQUATIONS  453 

1.  The  sign  of  the  first  term  is  not  changed. 

2.  The  second  sign  of  any  variation  is  not  changed. 

3.  The  second  sign  of  any  permanence  becomes  ambiguous. 

4.  There  is  added  one  term  at  the  end  whose  sign  is  opposite 
to  the  preceding  one. 

Hence  the  only  signs  which  may  be  changed  are  the  second 
terms  of  the  permanences.  But  a  change  in  the  last  term  of 
the  permanences  cannot  decrease  the  number  of  variations. 

Considering  the  additional  variation  at  the  end,  it  is  evident 
that  the  product  must  contain  at  least  one  variation  more  than 
the  original  expression. 

Hence  the  total  number  of  positive  roots  cannot  be  greater 
than  the  number  of  variations. 

536.  To  prove  the  theorem  for  negative  roots,  consider  the 
equation  ^^^^^0,  .  (1) 

and  change  the  signs  of  all  roots  of  (1)  by  substituting  —x 
for  X. 

I.e.  f{-x)=0.  (2) 

Since  (2)  is  obtained  from  (1)  by  changing  the  signs  of  the 
even  terms,  the  number  of  permanences  in  (1)  equals  the  num- 
ber of  variations  in  (2). 

But  (2)  cannot  have  more  positive  roots  than  it  has  varia- 
tions. Hence  (1)  cannot  have  more  negative  roots  than  it  has 
permanences. 

E.g.  ix^  —  2x^-63(^-\-7x^  +  2x-l  cannot  have  more  than  three 
p6sitive  and  two  negative  roots. 

537.  Incomplete  equations.  The  preceding  proof  refers  only 
to  complete  equations,  but  every  incomplete  equation  can  be 
completed  by  the  introduction  of  zero  coefficients. 

Thus,  a^— 4.r^—  3  a?  —  7  =  0  may  be  written 

a^±Oa^±  Ox*^  4:x'±0ar-3x^  7  =  0. 


454  ADVANCED  ALGEBRA 

The  smallest  number  of  variations  is  obtained  by  making  tbe 
signs  of  the  zero  coefficients  agree  with  the  preceding  terms,  viz. 

+  1  +  0  +  0-4-0-3-7. 

Evidently  the  number  of  variations  is  then  the  same  as  in 
the  original  equation,  and  hence  Descartes^  Bale  for  positive  roots 
may  be  applied  to  incomplete  equations. 

538.  For  negative  roots,  however,  the  number  of  perma- 
nences can  often  be  made  smaller  by  the  introduction  of  zero 
coefficients.  E.g.  the  above  equation  has  two  permanences. 
But  writing  the  coefficients  as  follows : 

1-0  +  0-4  +  0-3-7, 

the   equation  has  only  one  permanence. 

Hence  Descartes^  Rule  for  negative  roots  should  not  be  applied 
directly  to^  incomplete  equations.  The  most  convenient  method 
in  such  a  case  is  to  transform  the  equation  into  another  one 
whose  roots  have  opposite  signs,  as  illustrated  in  the  next 
article. 

539.  In  many  incomplete  equations,  the  greatest  possible 
number  of  real  roots  does  not  equal  the  degree  of  the  equation, 
hence  in  many  cases  imaginary  roots  'can  be  detected  by  the 
rule. 

Ex.  Prove  that  a^  +  3a^  —  5a;+l  =  0  has  at  least  four 
imaginary  roots. 

As  the  equation  has  two  variations,  it  cannot  have  more  than  two 
positive  roots. 

Substituting  —  x  for  a;, 

x6  +  3  a;2  +  5  X  +  1  =  0. 

Since  this  equation  has  no  variations,  it  cannot  have  any  positive  root, 
and  hence  the  original  equation  cannot  have  any  negative  roots.  That  is, 
the  total  number  of  real  roots  cannot  exceed  two. 

Therefore  the  given  equation  has  at  least  four  imaginary  roots. 


THEORY  OF  EQUATIONS  455 

540.    It  follows  from  Descartes'  Rule  that : 

1.  If  all  signs  of  an  equation  are  positive,  the  equation  has  no 
positive  roots. 

2.  If  the  signs  of  a  complete  equation  are  alternately  positive 
and  negative,  the  equation  has  no  negative  roots. 

EXERCISE  163  • 

Apply  Descartes'  Rule  to  the  following  equations : 

1.  a:«-2a;*-3a^-4iB2-2a;  +  l  =  0. 

2.  a;«-5ar*  +  2a;*-3a^  +  2«2_3a._7^()^ 

3.  5a;«-6x^  +  2x^4-3aj3  +  5i»2  4.2ic  +  l  =  0. 

4.  6««  +  a^  +  7a;^  +  9a^-2a^-5a;-6  =  0. 

All  the  roots  of  the  following  equations  are  real  j  determine 
their  signs  : 

5.  8a^  +  12a^-4a;-l  =  0. 

6.  27a.-3-54a^+25a;  +  l=0. 

7.  2aj3-5a^-13a;  +  30  =  0. 

8.  a;^-2a^-lla^-f6ic4-2  =  0. 

9.  4.x^-4.i»?-lS:x^  +  l^x-&=zO. 

10.  4  a^  -  41  aj^  +  6  ar^  +  73  a;  +  30  =  0. 

Determine  the  least  possible  number  of  imaginary  roots  in 
each  of  the  following  equations  : 

11.  o^J^2x^-^x-2  =  0. 

12.  a;«-3a:3  +  2a;-9  =  0. 

13.  a;7  +  2a^-7a:2_9^0. 

14.  a:«  +  2ar'-2a;-7  =  0. 


456  ADVANCED  ALGEBRA 

15.  a^-}-2a:^-3x*-^2x-\-l  =  0. 

16.  3a;'-5a;*  +  4a^-5  =  a 

17.  a:i«-l  =  0. 

18.  a;^3-l=0. 

19.  aj**  rf  1  =  0,  if  n  is  even. 

20.  a;«  +  2a;^  +  2a^  +  5  =  0. 

21.  aaj^  +  6x  +  c  =  0,  if  a,  6,  and  c  are  positive. 

LOCATION  OF  ROOTS 

541.  Limits  of  roots.  The  solution  of  numerical  equations 
may  often  be  simplified  by  determining  between  what  limits 
the  roots  lie. 

542.  A  superior  limit  to  the  real  roots  is  a  number  greater 
than  the  greatest  root. 

543.  An  inferior  limit  to  the  real  roots  is  a  number  smaller 
than  the  smallest  root. 

544.  A  superior  limit  may  be  obtained  by  grouping  the  terms 
so  that  each  group  contains  not  more  than  one  negative  term, 
and  determining  a  value  of  x  which  makes  each  group  positive. 

Ex.  1.   Find  a  superior  limit  to  the  roots  of  the  equation 

x*  +  4:a^-6x'  +  2^x-e0  =  0. 
Grouping  terms  and  factoring, 

x%x^  _  6)  +  4  (x3  -  15)  +  24  X  =  0. 

Evidently  each  term  is  positive,  if  x  ^  2^. 
Hence  2^  is  a  superior  limit. 


THEORY  OF  EQUATIONS  457 

Ex.  2.    Find  a  superior  limit  to  the  roots  of  the  equation 

Multiplying  by  3,         3  a;^  _  6  a:2  -  9  a;  -  48  =  0. 

Grouping,  (x«  -  6  x^)  +  (x'  -  9  x)  +  (x«  -  48)  =  0. 

Or  a;2(5c-6)+a;(x2-9)  +  (x«-48)  =  0. 

Evidently  no  term  is  negative  if  a;  ^  6. 
Hence  6  is  a  superior  limit. 

Since  the  limits  obtained  from  the  three  groups  differ  considerably,  we 
modify  the  method  as  follows  : 

Multiplying  the  given  equation  by  4,  and  grouping, 

(2  x3  -  8  x2)  +  (x3  -  12  X)  4-  (x3  -  64)  =  0. 

Factoring,        2  x2(x  _  4)  +  x  (x2  -  12)  +  (x^  -  64)  =  0. 

No  terra  is  negative,  if  x  ^  4. 
Hence  4  is  a  superior  limit. 

545.  To  determine  an  inferior  limit,  change  the  signs  of  all 
roots  by  the  transformation  of  §  531,  and  determine  the 
superior  limit. 

Ex.  3.   Find  an  inferior  limit  to  the  roots  of  the  equation 

Let  X  =  -  y. 

Then  y^ -20y^  -  i8y -{■  120  =  0  (§  531).  (1) 

Multiplying  by  2,  and  grouping,  (y*  -  40  y^)  +  (y*  -  96  y)  +  240  =  0. 

Factoring,  y'^(y^  -  40)  +  y(y^  -  06)  +  240  =  0. 

I.e.  6\  is  a  superior  limit  to  y. 

Therefore  —  6^  is  an  inferior  limit  to  x. 

A  closer  limit  is  obtained  by  multiplying  (1)  by  3. 

2yi-60y2  +  yi  -  Uiy  +  360  =  0. 
2y2(y2  _  30)  4-  y(yS  -  144)  +  360  =  0. 

I.e.  6^  is  a  superior  limit  to  y. 
Therefore  —  5^  is  an  inferior  limit  to  x. 


458 


ADVANCED  ALGEBRA 


EXERCISE   164 

Determine  a  superior  and  an  inferior  limit  to  the  roots  of  the 
following  equations : 

2.  x*-]-3x^-4:X-14:  =  0.  5.    2x^-x^-2x'  +  x-4:  =  0, 

3.  3a^-5a^  +  3a;-9  =  0.  6.    Sa^ -4.x^  -  5x-7  =  0. 

7.  4.x^-{-2x^-{-2x^-7x  +  2^  =  0. 

8.  5x'-{-2x*-3x^-i-2x^~x-AS  =  0, 

9.  6x^-\-5x^-\-4:a^-\-3x^  +  2x  +  l  =  0. 
10.  a;«  -  2  aj2- 12600  =  0. 


546.  A  function  f(x)  is  continuous  if,  for  any  value  of  x,  an 
intinitesinial  change  in  the  independent  variable  x  produces  an 
infinitesimal  change  in  the  function,  or  if 

f{x-{-h)-f(x)-^0,iih  =  0. 

Thus,  x^  is  continuous  for  any  value  of  x,  as  (x  -\-  hy  —  x^  =  2  hx  +  h'^, 
a  quantity  which  for  any  finite  x  approaches  zero  as  a  limit,  if  ^  =  0. 

547.  The  meaning  of  continuity  is  well  illustrated  by  repre- 
senting functions  graphically.  The  line  FF',  which  represents 
the  graph  of  a  function 
f(x),  is  supposed  to  be  a 
continuous  line,  i.e.  a  line 
without  a  break.  Such 
a  function  is  a  continuous 
function,  and  the  inspec- 
tion of  the  diagram  shows 
that  any  increase  (h  or 
AC)  in  the  independent 
variable  and  the  corresponding  increase  (cB)  of  the  function 
simultaneously  approach  zero  as  a  limit. 


C 


ThEOEY  OF  EQUATIONS 


459 


548.  On  the  other  hand,  the 
second  diagram  represents  a  dis- 
continuous function  (PF'QQ').  It 
is  apparent  that  /(2)  will  have  the 
finite  increase  of  +  1  if  2  increases 
by  an  infinitesimal  quantity  h. 

Hence  this  function  is  discon- 
tinuous. 


_  (f(^K 


549.    Tlie  function  ax"'  is  continuous  if  m  is  an  integer. 

For  a{x -\- hy  —  ax"^ 

=  a\j)if"  4-  mar-Vi  -h  "^0^^^-%^  -\-  -^  -  iC*] 

=  a  '  h[mxr-^  4-  '^C^'^-%  4  •-  1- 

If  ^  =  0,  the  right  member  =  0,  for  any  finite  x. 
Hence  ace"'  is  continuous 


550.  The  function  f(x)  —  a(pf  -\-  aiOif'^  -{-  -  ^  -  -{-  a^  is  contimious. 
Any  infinitesimal  increase  of  h  produces  an  infinitesimal 

increase  of  each  term,  and  therefore  of  the  whole  function 
(§  412,  1). 

Hence  f(x)  is  continuous. 

551.  If  f(a)  and  f(h)  have  opposite  signs,  at  least  one  root 
must  lie  between  a  and  b. 

If  X  changes  gradually  from  a  to  6,  f{x)  changes  gradually 
from  /(a)  to  f{h),  i.e.  from  a  positive  to  a  negative,  or  from 
a  negative  to  a  positive,  value. 
Hence  it  must  pass  at  least  once 
through  zero,  i.e.  f(x)  has  at 
least  one  root  between  a  and  b. 

552.  The  preceding  proposi- 
tion becomes  obvious  by  inspec- 
tion of  the  graph.     Evidently  a 


'f(^) 


460 


ADVANCED  ALGEBRA 


continuous  line  cannot  join  two  points,  A  and  B,  on  opposite 
sides  of  OX  without  intersecting  OX  at  least  once. 

There  may,  however,  be  three,  five,  or  any  odd  number  of 
roots,  as  appears  from  the  next  diagram. 


L 


/2\ 


■>x 


553.  The  preceding  article  furnishes  the  principal  method 
for  locating  incommensurable  roots. 

Ex.  1.    Show  that  the  equation 

a;4  +  5  ^  -  60  £c2  ^  70  a;  +  100  =  0 

has  a  root  between  4  and  5. 

1  +    5  _  60  +  70    +  100  1_4 

-I-    4  4-  36  -  96    -  104 
1  _}.    9  _  24  -  26  ;  -      4.     I.e.  /(4)  =  -  4. 

1  +    5  _  60  +  70    +  100  |_5_ 

+    5  +  50-50    +100 
1  +  10  -  10  +  20 ;  +  200.     I.e.  /(5)  =  +  200. 

Hence  at  least  one  root  lies  between  4  and  5. 

554.  If  x==cc,  then  f(x)~-{- CO. 

If  x  =  —  ao,  then  f(x)  =  +  oo  if  n  is  even,  and  f(x)  =  —  cc  if 
n  is  odd. 

For  it  can  be  shown  that,  for  a  very  large  value  of  x,  the  first 
term  of  the  function  is  greater  than  the  sum  of  the  remaining 
terms,  and  hence  the  function  approaches  the  values  -f  oo  or 

—  CO. 


THEORY  OF  EQUATIONS  461 

555.  Every  equation  of  an  odd  degree  has  at  leobst  one  real  root 

whose  sign  is  opposite  to  that  of  the  last  term. 

If  X  equals  respectively        —  oo,    0,  +  oo, 
then /(a;)  equals  respectively  —  oo,  p^,  H-oo. 

Hence  if  p^  is  positive,  there  must  be  at  least  one  root  between 
—  X  and  0,  i.e.  there  must  be  a  negative  root. 
Similarly  if  p„  is  negative,  there  must  be  a  positive  root. 
Thus,  a;^  +  a;2  _j.  X  -f  1  has  at  least  one  negative  real  root. 

556.  Every  equation  of  an  even  degree  and  a  negative  absolute 
term  (p„)  has  at  least  one  positive  and  one  negative  root. 

If  X  equals  respectively        —  oo,    0,  -f-  oo, 

f(x)  equals  respectively  H- ^,  — ,  +00. 

Hence  there  must  be  at  least  one  positive  and  one  negative 
root. 

Thus,  xs  +  4ac2  —  2aj  —  7  =  0  has  at  least  one  positive  and  one  negative 
root. 

Ex.  2.    Locate  the  real  roots  of  the  equation 
a;4_i0a^  +  33i/-40a;4-14  =  0. 

Since  there  are  no  permanences,  there  can  be  no  negative  roots,  and 
the  method  of  §  544  shows  that  10  is  a  superior  limit. 

Evidently  /(O)  =  14,  and  /(I)  =  —  2.  By  synthetic  division  we  obtain 
/(2)=  2,  f(J\)=  2,  /(4)  =  -  2,    and  /(6)  =+14. 

Hence  the  four  roots  are  incommensurable  and  lie  respectively  between 
0  and  1,  between  1  and  2,  between  3  and  4,  and  between  4  and  5. 

Ex.  3.  Find  the  nature  of  the  roots,  and  locate  the  real  roots 
of  the  equation :  a^-2a^-6  =  0. 

Since  the  degree  of  the  equation  is  even,  and  the  absolute  term  is 
negative,  there  must  be  one  positive  and  one  negative  root  (§  550). 
Descartes'  Rule  shows  that  there  can  be  no  more  than  one  positive  and 
one  negative  root. 


462  ADVANCED  ALGEBRA 

Hence  there  must  be  one  positive,  one  negative,  and  two  imaginary- 
roots. 

Since  x^{x^  —  4)  +  (a;*  —  12)  is  positive  if  aj  =  2,  we  liave  a  superior 
limit  2  ;  and  similarly  obtain  the  inferior  limit  —  2. 

Hence  we  have  to  find  the  sign  of  /(-2),  /(-  1),  /(O),  /(I),  and 
/(2).  A  simple  calculation  shows  that  the  required  five  signs  are 
respectively  H (-  • 

Therefore  one  root  lies  between  —  2  and  —  1,  another  between  1  and  2. 


EXERCISE  165 

Determine  the  character  of  the  roots  of  the  following  equa- 
tions : 

1.   2a;*  +  22a^  +  5a;-ll  =  0. 

3.  i»4-3ar^-12a;-16  =  0. 

4.  a^  -\-ax  —  b  =  0,  if  a  and  b  are  positive. 

5.  a^  -\-  ax  -\-  b  =  0,  ii  a  and  b  are  positive. 

Locate  the  roots  of  the  following  equations : 

6.  4ic3-10a^  +  2a;-}-3  =  0.     •      9.   ar^ -f4a;-5  =  a 

7.  a^  +  4:x'-4x-S=0.  10:  a^-f2a;  +  2  =  0. 

8.  aj3-6ic2  +  4a;  +  10  =  0.  11.   a;^-5a^-4aj  +  19  =  0. 


*    CHAPTER   XXIX 
SOLUTION  OF  HIGHER  EQUATIONS 
COMMENSURABLE  ROOTS 

557.  The  principal  features  of  the  method  for  finding  com- 
mensurable roots  were  discussed  in  the  precoding  chapter. 
There  are,  however,  a  few  additional  propositions,  which  greatly 
facilitate  the  finding  of  commensurable  roots. 

558.  If  r  is  an  integral   root  oj  the   equation  with   integral 

coefficients  f(x)  =  0,  then  J-^-1-  must  he  an  integer  for  any  iiitegral 
X     r 

value  of  x^  for  this  quotient  is  a  rational,  integral  function  of  x. 

Since  /(I)  is  easily  found,  we  usually  apply  this  test  first 
for  x—\.     x.e,  "  ^^^    must  he  an  integer  for  any  integral  root  r. 

Similarly  for  -^^"~   ^  ,  etc. . 

Ex.  1.    Determine  the  integral  roots  of  the  equation 

.  The  factors  of  -  20  are  ±  1,   ±2,   ±4,   ±5,   ±  10,   ±  20. 
/"(l)  =  ~  7  ;  that  is,  1  is  not  a  root. 

If       "  =    2,     ^,     ^,    ;0,     ;zp,  _  ;,  -  ^,  _  ^,  -  ^,  _  ;^,  -  ;z0, 

then    1  -    -..=:- 1,  -3,  -;!,-  R,  -;?>,      ?!,       3,      ^,      ^,      %h      n^ 

Reiecting  all  values  of  1  —  r  which  are  not  factors  of  —  7,  and  the 
corresponding  vahies  of  r,  only  2  remains. 

As  <vnthetic  division  shows  that  2  is  not  a  root,  the  equation  has  no 
integral  roots 

463 


464  ADVANCED  ALGEBRA 

Ex.  2.   Determine  the  integral  roots  of  the  equation 

2a;*_49ar'  +  281a;2  +  72a;-180-0. 

The  inferior  and  superior  limits  are  respectively  —  2  and  +  25.  Hence 
the  roots  may  be  —  1,   +  1,  2,  •••  20. 

/(I)  =  2  -  49  +  281  +  72  -  180  =  126,  hence  1  is  not  a  root. 

If        r=-l,      2,      3,      4,      ^,      ^,      ^,      10,      ;^,      15,      %f>,      %% 

then  l-r=     2,  -1,  -2,  -3,  -^,  -^,  -^,  -  9,  -;;,  -14,  -JJ,  -;p. 

Eejecting  all  values  of   1  —  r  which  are  not  factors   of   126,  and  the 
corresponding  values  of  r,  there  remain  —1,  2,  3,  4,   10,  and  15. 
/(-  1)  =  2 +  49 +  281 -72 -180  =  80;  hence  -1  is  not  a  root. 

If  T=     %       3,       4,       m       15, 

_l-r  =  -^,   -4,   -5,   -Ih   -16. 

Rejecting  all  values  of  —  1  —  r,  which  are  not  factors  of  80,  and  the 
corresponding  values  of  r,  there  remain  3,  4,  and  15  as  possible  roots. 

Synthetic  division  shows  that  3  and  4  are  not  roots,  but  15  is  a  root. 
Hence  15  is  the  only  integral  root. 

559.  Newton's  method  of  divisors  is  very  similar  to  synthetic 
division;  but  the  numerical  work  is  sometimes  simpler,  espe- 
cially when  the  exponents  or  the  coefficients  are  very  large. 

Let  r  be  a  root  of  the  equation 

a^x''  +-  a^x""-^  +-  otg^""^  +  '  •  •  +  drx-Y^  +  a„  =  0. 
Substituting,  dividing  by  r,  and  transposing, 

—  +  «n-i  =  —  cfn-2^  —  ...  —  a^r'^-'^  —  aor**"^. 

T 

The  right  member  being  an  integer,  the  left  member  must 
be  an  integer. 

Denoting  this  integer  by  q^,  dividing  by  r,  and  transposing, 


SOLUTION  OF  HIGHER  EQUATIONS  465 

As  before,  the  left  member  must  be  an  integer.  Denoting  it 
by  Qi,  dividing  by  ?-,  and  transposing, 

7  +  a«-3  =  -  ««-4^ ^i^""*  -  cio^^'^- 

Continuing  this  process,  we  obtain  finally 

Qn-l    , 

-;r-  +  «i  =  -«o^. 

y  +  «o  =  0. 

560.  The 'work  can  be  arranged  in  a  manner  very  similar  to 
the  one  used  for  synthetic  division.  In  fact,  the  work  differs 
from  synthetic  division  only  in  two  points  : 

(1)  We  commence  at  the  last  term  and  work  toward  the  first. 

(2)  We  divide  instead  of  multiplying. 

Ex.  Determine  if  2  is  a  root  of 

2  -  1  -  10  4- 15  -  14  [2 

_2_34-    4-    7 


0-4-   6+    8-14 


Explanation.  Take  down  — 14,  divide  it  by  2,  and  add  the  quotient 
(—7)  to  the  preceding  coefficient.  Divide  this  sum  (8)  by  2,  and  add 
the  quotient  to  the  preceding  coefficient,  etc.  As  there  is  no  fractional 
quotient,  and  as  the  last  sum  is  zero,  2  is  a  root.  The  coefficients  of  the 
depressed  equation  are  the  numbers  in  the  second  line  with  their  signs 
changed,  viz.   +2  +  3-4  +  7. 

561.  If  a  number  is  not  a  root,  this  fact  is  discovered  as  soon 
as  we  come  to  a  fractional  quotient.  This  constitutes  the 
advantage  of  Newton's  method  over  synthetic  division. 

I.e.  to  determine  if  7  is  a  root  of  xHa:^-10x^+15a;-14=0,  we  have 

1  +  1_10  +  15-14[7 

-   2 

+  13-14 
Hence  7  is  not  a  root. 


466  ADVANCED  ALGEBRA 

562.  The  method  for  finding  commensurable  roots  may  be 
summarized  as  follows  : 

1.  Make  the  coefficients  integers,  and  determine  all  integers  that 
may  be  roots.  To  be  roots,  integers  must  satisfy  the  following 
conditions  : 

(a)   They  must  be  factors  of  the  absolute  term. 

(6)    Tfiey  must  lie  betweeii  the  inferior  and  the  superior  limits. 

(c)  They  must  conform  with  Descartes'  Rule. 

m) 

(d)  •      ^    must  be  an  integer,  if  r  is  an  integral  root.     Simi- 

larly  for  4^,  ^,  etc. 

2.  Try  if  the  numbers  which  satisfy  (1)  are  roots  of  the  equa- 
tion.    The  trial  may  be  made  by 

(a)  Synthetic  Division. 

(b)  Newton^ s  Method. 

Note.  Very  small  numbers,  as  +  1,  or  —  1,  are  more  easily  tested  by 
direct  substitution.     E.g.  if  x  =  1,  then  /(x)  =  the  sum  of  the  coefficients. 

3.  After  all  integral  roots  are  removed,  the  equation  has  no 
more  commensurable  roots,  if  the  first  coefficient  is  unity. 

4.  If  the  first  coefficient  is  not  unity,  transform  the  equation 
into  one  in  its  simplest  form  (i.e.  first  coefficient =1)2*??'^^  Integral 
coefficients,  and  proceed  as  before. 

Note.  If  the  first  coefficient  is  very  small,  it  is  sometimes  convenient 
to  apply  this  last  transformation  (4)  at  the  very  beginning. 

Ex.  1.  Solve  the  equation 

The  inferior  and  superior  limits  are  respectively  —  11  and  8. 
Hence  the  roots  may  be  ±1,   ±2,  ±3,   ±4,   ±5,   ±6,   -  8,   - 10. 

/(I)  =  2  +  5  -  114  -  238  -  240  =  -  585.     Hence  1  is  not  a  root. 

If    r=-;p,  -8,  -^,  -^,  -4,  -3,  -2,  -;,      2,      3,      4,      ^,      6, 

l-r=    XX,      9,      I     ^,      5,      ^      3,      ^  -1,  -^,  -3,  -^,  -5. 


SOLUTION  OF  HIGHER  EQUATIONS  467 

W.ejecting  all  values  of  1  —  r  which  are  not  factors  of  —  586,  there 
H'uain  the  following  values  of  r,  —8,   —4,   -2,  2,  4,  6. 

/(  -  1  )  =  2  -  5  -  114  +  238  -  240  =  -  119. 

If  r  =  -8,   -^,   -2,       ^,       -I,       6, 

then  -l-r=     7,       3,       1,-3,-^,-7. 

Rejecting  all  values  of  —  1  —  r,  which  are  not  factors  of  — 119,  we 
have  r=:-8,   -2,  or  +6. 

Testing  cc  =  —  2,  i.e.  dividing  by  -oc  +  2,  we  obtain  the  remainder  —  228  j 
that  is,  —  2  is  not  a  root,  and  /(  —  2)  =  —  228. 

If  r  =  -8,   4-^, 

then  _2-r=     6,   -^. 

But  —  8  is  not  a  factor  of  —  228,  hence  the  only  possible  inteicral  root 
is  the  one  corresponding  with  6,  i.  e.  —  8. 

Dividing,  2+    5- 114 -238 -240  | -8 

-16+    88  +  208  +  240 
2-11-26-30         0 

I.e.  a;  =  —  8  is  the  only  integral  root,  and  2a;8  — lla;2  — 26a;^30s=0 
is  the  depressed  equation. 

Dividing  by  2,  x^  -  V  a;2  -  13  x  -  15  =  0. 

Let  x  =  ^,y^-Uy'^-b2y- 120  =  0. 

The  limits  to  y  are  respectively  —  7|  and  +17. 

Rejecting  all  multiples  of  2,  since  x  cannot  be  an  integer,  we  have  the 
possible  roots  ±1,  ±3,  ±5,  + 16. 

/(I)  =  1  -  11  -  52  -  120  =  -  182.     Hence  1  is  not  a  root. 

If  r  =  -^,   -3,   -1,       3,       ^,       16, 

\-r=     ^,       ^,       2,   -2,   -i,  -14. 

I.e.   —1,3,  16,  are  the  only  integers  that  may  be  roots. 

/(-  1)  =  -  1  -  11  +  52  -  120  =  -  80.     Hence  -  1  is  not  a  root 

Synthetic  division  proves  that  3  is  not  a  root,  while  15  is  a  root. 
The  depressed  equation  is  y^-\-Ay  +  ^  =  0.     Solving  by  formula, 

y  =  -2±2i. 

Therefore  y  =  -  2  +  2  i,   -  2  -  2 1,  or  16. 

Hence  the  required  roots  are  -  8,  J^,  -  1  + 1,  -  1  —  <. 


1G8  ADVANCED  ALGEBRA 

Ex.  2.     Solve  by  Newton's  method, 

8  ic*  -  68  ar^  +  190  0^2  _  199  .p  _j.  gQ  ^  Q^ 

The  equation  can  have  positive  roots  only,  and  8|  is  a  superior  limit. 
Hence  we  have  to  try  1,  2,  3,  4,  5,  6. 

/(I)  =  8  -  68  +  190  -  199  +  60  =  -  9.     Hence  1  is  not  a  root. 

K  r=     2,       ^,       4,       ^,       ^, 

l-r  =  -l,  -^,  -3,  -^,  -^. 

Rejecting  all  values  of  1  —  r  which  are  not  factors  of  —  9,  there  remain 
r  =  2,  or  4. 

...4.190-199  +  6012 
+   30 

- 169  +  60 

8_68  +  190-199  +  60(4 
-8  +  36-46+15 

0  _  32  +  144  -  184  +  60 

Hence  4  is  a  root,  but  2  is  not  a  root. 

As  the  only  possible  integral  root  was  removed,  the  depressed  equation 

8  a;3  —  36  a;2  +  46  a;  —  15  =  0,  can  have  no  integral  roots. 
Dividing  by  8,  x^  -  ^  x^  +  \^x-  J/-  =  0. 

Let     x  =  l,     then    y3  -  9^/2  +  23?/ -  15  =  0. 

The  possible  integral  roots  are  1,  3,  5. 

/(I)  =1-9  +  23-15  =  0. 
Hence  we  have  y  =  1,  and,  by  division,  y^  _  s  j/  +  16  =  0. 
Or,  (y_3)(y-5)=0. 

That  is,  2/  =  1,  3,  or  5. 

Hence  the  required  four  roots  are  4,  ^,  f ,  f. 


SOLUTION  OF  HIGHER  EQUATIONS  469 

EXERCISE  168 

Find  all  commensurable,  or,  if  possible,  all  roots  of  the  fol- 
lowing equations : 

1.  ar'-49a;-120  =  0. 

2.  a^-Uaf  +  STx-llO^zO. 

3.  a;*-10ic3  +  35aj2-50a;  +  24  =  0. 

4.  aj*-45ar'-40.T  +  84  =  0. 

5.  8a^  +  34a;--79a;  +  30  =  0. 

6.  3iB»-8a;^  +  3a5-t-2==0. 

7.  4a;3-3a^-25a;-6  =  0. 

8.  2a:»-15a;2  +  31x-12  =  0. 

9.  2a^-.13ar^  +  13a;4-10  =  0. 

10.  3ic3  4-16x2  +  23cc  +  6  =  0. 

11.  a^_i^4^_|_7a;_a^  =  0. 

12.  a^-Ux'-5x  +  5S  =  0. 

13.  a^-13x^  +  A9x  -4:5  =  0. 

14.  x*+a^-24a^  +  43a;-21=0.  ■ 

15.  x*-{-8a^-lSx^-92x-h96  =  0, 

16.  24a^-26cc2  +  9a;-l  =  0. 

17.  2a^-llar'4-17aj~6  =  0. 

18.  a;*  +  19a;3  +  123a;2^3Q5^_^200  =  0. 

19.  2a;*-lla^  +  16jc2-a;-6  =  0. 

20.  3a;^-20aj3  +  41iK2-20a;-12  =  0. 

21.  3a;*-19a^  +  39a;2_29a;4-6  =  0. 

22.  6aj*-13a^H-20a;^-37a;  +  24  =  0. 

23.  a;^  4- 29  a^  +  287  a;2  + 1147  aj  + 1560  =  0. 

24.  »«-3a5*-8aj3  +  24a^-9a;  +  27  =  0. 


470 


ADVANCED  ALGEBRA 


INCOMMENSURABLE   ROOTS 

563.  Graphic  method.  The  simplest  method  for  determining 
the  incommensurable  roots  of  an  equation  is  the  graphic 
method. 

The  essential  features  of  this  method  were  discussed  in 
Chapter  XVII.  The  theory  of  equations,  however,  sometimes 
simplifies  the  work.  Thus,  we  may  use  synthetic  division  for 
finding  the  various  values  of  a  function,  we  may  determine  the 
limits  to  the  roots,  we  may  transform  the  equation,  etc. 

Ex.    Find  graphically  the  roots  of  the  equation 

24a^-26a;2  +  9a;-l  =  0. 

By  Descartes'  Rule  there  are  no  negative  roots,  but  there  must  be  at 
least  one  positive  real  root  (§  555).     The  superior  limit  is  lj\. 

To  avoid  fractions,  we  multiply  the  roots  by  10;  i.e.  make  ?/=  lOic, 


or, 


24  y3  -  260 1/2  +  900  2/  -  1000  =  0. 

Dividing  by  4,  6  y^  _  55  yi  +  225  y  -  250  =  0. 

/(O)  =  -  250,  /(I)  =  6  -  65  +  225  -  250  =  -  84. 

By  synthetic  division  we  find 

/(2)  =  -  12,  /(3)  =  +  2,  /(4)  =  -  6,  /(5)  =  0,  /(6)  =  56. 

Locating  the  points 

(2, -12),  (3,  2),  (4, -6),  (5,0),  (6,  56), 

and  joining,  produces  the  gi-aph  ABG, 
vv^hich  intersects  the  ic-axis  in  P,  P', 
and  P".  By  measuring  OP  and  OP'., 
we  obtain  the  approximate  roots  2.5 
and  3.3,  while  5  is  an  exact  root. 

Hence  the  lequired  roots  are  .25,  .33, 
and  .5. 


Y' 

30 
•20 

/ 

r 

/ 

0 

?^ 

k^' 

/p" 

X 

X' 

0 

/ 

^ 

Ly 

■  ' 

/ 

30 

/ 

A 

SOLUTION  OF  HIGHER  EQUATIONS  471 

Note.     To  find  more  exact  values  of  x,  the  portion  of  the  diagram 
which  contains  Pi  or  P^  should  be  drawn  on  a  larger  scale  (see  page  279). 


EXERCISE    167 

Find  graphically  the  roots  of  the  following  equations  : 

1.  a^  +  8a^  +  19a;  +  13  =  0. 

2.  a;^4-4ic3  +  18  =  0, 

3.  a^-13iB2  +  38a;  +  17  =  0. 

4.  100a^  +  4a;-l  =  0. 

5.  64:a^-16a^-5x  +  :^  =  0. 

U  f(x)  =  x*-6o^-\-7a^-\-6x-4:. 

6.  Find  the  approximate  values  of  the  roots  of  f(x)  =  0. 

7.  Solve /(ar)  =  4.      . 

8.  Determine  the  number  of  real  roots  of  the  equations 

f{x)=:l,f{x)  =  100,f(x)  =  -50. 


9.    Solve  the  system :  . 

f  y  =  f(x), 

10.  Solve  the  system  :  \^  ' 

If /(a;)=ar'-a^-llaj3  +  9ar^  +  18a?-4. 

11.  Solve /(«)  =  0. 

12.  Solve  /(a?)  =  —  4. 

13.  Determine  the  number  of  real  and  imaginary  roots  of 
fix)  =  -  20,  fix)  =  5,  f{x)  =  20,  fix)  =  36,  f{x)  =  100. 


14.    Solve  the  system :  (         J     }' 


472  ADVANCED  ALGEBRA 

Solve : 

15.    af  =  10.  17 


y  =  a^-2x-\-2,  (  A x^ -\- y' =  4:, 


16 

t  aj  +  3/  =  4. 

19 


1^  +  2/2^8, 


,0^  +  2/2=8 

20.   Has  the  system  \  ^  any  real  roots  ? 

'     x+y=S      ^ 


564.    Roots  diminished  by  a  given  number. 
To  transform  an  equation  into  another  one  zvhose  roots  are  less 
by  h  than  the  roots  of  the  given  equation. 

Let  ao^"  +  cLi^'"'^  +  CL^~^  H h  a^-iX  ■\'a^  =  <)  (1) 

be  the  given  equation. 

Make  y  =  x  —  h,  then  x  =  y-\-  h. 
Substituting  in  (1), 

«oCv  +  J^T  +  <y  +  ^0""'  +  -  +  «n-i(2/  +  h)  4-  «n  =  0.      (2) 

If  we  sliouk]  expand  all  parentheses,  and  collect  equal  powers 
of  y,  we  should  obtain  an  equation  of  the  form 

M"  +  gi3/"-'  +  922/"-'  +  -  +  gn  =  0.  (3) 

The  work  of  expanding  the  parentheses  is,  however,  very 

tedious.     It  is  simpler  to  find  the  coefficients  %  Qd  ^2  ••*  ^n  by 

the  following  consideration. 

Since  y  =  x  —  h,  equation  (3)  may  be  written 

q^(x  -  hy  +  q^(x  -  hy-^  +  q2(x  -  hy-'  + . .  . 

+  qn-i{x-h)  +  q,,  =  0.  (4) 

But  this  equation  is  the  original  equation  (1)  written  in  a 
different  form.  Therefore  any  division  applied  to  (4)  will  give 
the  same  quotient  and  the  same  remainder  as  if  applied  to  (1). 


SOLUTION  OF  HIGHER  EQUATIONS  473 

But  the  coefficients  q^,  q^,  •••  can  be  obtained  from  (4)  and 
hence  from  (1)  by  the  following  operations: 

Divide  (4)  by  x  —  h,  then 

Quotient  =  qo{x  -  hy-^  +  qiix-  hy-^  +  •  •  •  +  5'n-i  5 

Remainder  =  g^. 

Dividing  the  last  quotient  by  x  —  ^,  we  have 

Qnotient  =  qo(x-hy-'  +  q,{x-hy-'-{-  ••.  +9n-2; 

Remainder  =  g„_i. 

Evidently,  by  successive  division  of  the  quotients  by  x  —  h, 
we  should  obtain  the  coefficients  5„  •••  ^i  as  remainders,  while 

go  =  cio. 

565.  To  diminish  the  roots  of  an  equation  f(x)  =  0  by  h,  divide 
f(x)  by  x  —  h  until  the  remainder  does  not  involve  x.  This  re- 
mainder vnll  be  the  last  coefficient  (q^)  of  the  required  equation. 
The  remainder  obtained  by  dividing  the  quotient  by  x  —  h  is  the 
coefficient  g„_i,  etc. 

Ex.  1.  Find  an  equation  whose  roots  are  less  by  2  than  the 
roots  of  the  equation  3  a;*  -  20  a^  -h  5  ar^  -f  13  a;  +  70  =  0. 

The  division  may  be  arranged  as  follows  : 

3     -20     4-5     +    13  4-70  [2 
+   6     -28     -46-66 


3     _  14     -  23 
4-6-16 

-  33 

-  78 

4-  4 

3     -    8     -39 

+    6-4 

-111 

3-2 

+   6 

-43 

3+4 

Hence  the  required  equatio 

3x*  +  4xa 

nig 
-43a;2 

-llla;  +  4  =  0. 

i74  ,  ADV^NoUD    ALGEBRA 

HJx.  2     bind  aii  equation  wliose  roots  are  greater  by  3  than 
clie  rootb  ot  cne  equation  2x'^  —  3x^-{-5x  —  7  =  0. 

ifividently,  /t  =  —  3. 

3 


2 

+  0-3 
-6  +  18 

+  5 
-  45 

-  7 
+  120 

2 

-6+15 
-6+36 

-  40 
-153 

+  113 

2 

-12  +  51 

-6  +  54 

-193 

2      -  18  ,  +  105 


2     -24 
The  required  equation  is  2  x*  -  24  x^  +  105  x^  -  193  a;  +  113  =  0. 

566.  The  preceding  method  can  be  used  to  transform  an 
equation  into  another  one  in  which  the  second  term  is  wanting. 

Let  aoic"  +-  a^x""-^  -\ +  a„  =  0  (1) 

be  the  equation. 

Diminishing  the  roots  by  ^,  we  have 

y  =  x  —  h,OTX  =  y-{-h. 

Substituting  in  (1) 

aoiy  +  hy  +-  a,(y  +-  /i)""^  +...+-  a„  =  0. 

If  we  should  expand  and  combine  equal  powers  of  y,  we 
should  obtain  aonh  -\-  %  as  the  coefficient  of  y''~\ 
Making  this  coefficient  equal  to  zero,  we  obtain 

nao 

Hence,  if  we  diminish  the  roots  of  (1)  by -j  the  trans- 
formed equation  is  the  required  one.                      ^  ^ 

567.  If  the  equation  is  in  its  simplest  form  (i.e.  a^^l), 
diminish  the  roots  by  —  — ' ;  that  is,  by  —  ^  the  second  coeffi- 


SOLUTION  OF  HIGHER  EQUATIONS 


475 


cient  for  a  cubic,  by  —  J  the  second  coefficient  for  an  equation 
of  the  fourth  degree,  etc. 

Ex.  3.    Transform  the  equation 

into  another  one  whose  second  term  is  wanting. 

Diminish  the  roots  by  —  ^  of  —  2,  i.e.  .6. 

-2.       -6.         +3.  +2.  L^ 

.5     _    .75     -3.375     -    .1875 


1.5     -6.75 
.5     -    .5 


.375 
3.626 


+  1.8126 


1    -1. 

-7.25 

.5 

-    .25 

1     -    .5 

-7.50 

.5 

4.000 


0 
Hence  the  transformed  equation  is 

2/4 -7.5  2/2- 4^  +  1.8126  =  0. 


EXERCISE   168 

Transform  the  following  equations  into  others  whose  roots 
are  less  by  h : 

1.  2x'-4:a^-}-5x'-Sx  +  l  =  0,  h  =  2. 

2.  Sx'-19a^  +  22x^-17x-U  =  0,  h  =  l. 

3.  4a^-20a;2-17a;  +  12  =  0,  ^  =  3. 

4.  2.^•^-3ar^-9a;  +  71  =  0,  A  =  4. 

5.  ar'_6a;*  +  12a^-12  =  0,  h  =  5. 

6.  a;^-12a:»  +  7ar^-3a;  +  12  =  0,  ^  =  3. 

7.  ar^-l  =  0,  h  =  2. 

8.  a^4-aj4-}-ar^-l=0,  h  =  2. 

9.  iB«-2a;Hl  =  0,  A  =  1.5. 


476 


ADVANCED  ALGEBRA 


Transiorm  the  following  equations  into  others  whose  roots 
are  greater  by  k : 

10.  a;*  +  2a;2-17  3:4-12  =  0,  h  =  2, 

11.  a^-4:x'  +  3a^-}-2x'  +  Sx-^9  =  0,  h  =  l. 

Transform  the  following  equations  into  others,  in  which  the 
second  term  is  wanting : 

12.  aj*-4a^H-2a^-7  =  0.         15.    x^-^-Ax^ -7  x-{-2  =  0. 

13.  a^-9a^  +  5a;-3  =  0.  16.    x' -S x^ -\-2 x-7  =  0. 

14.  a^  +  6x^-3x-3  =  0.  17.    x^-2  a^-{-x'-x  +  l  =  0. 


568.   Horner's  method  of  approximation.    By  Horner's  method, 
incommensurable  roots  may  be  found  to  any  degree  of  accuracy. 
Let  it  be  required  to  find  the  positive  root  or  roots  of 

ar'_3a;_4  =  0.  '  (1) 

First  locate  the  roots  either  by  the  graphical  method  or  by 
§451. 

According  to  Descartes'  Rule,  the  equation  cannot  have  more 
than  one  positive  root,  and  by  §  555  it  must  have  one  positive 
root.     By  substitution  we  find 

/(O)  =  -  4,  /(I)  =  -  6,  /(2)  =  -  2,  /(3)  =  14. 

Hence  the  required  root  must  lie  between  2  and  3. 
Diminishing  the  roots  of  the  equation  by  2,  we  obtain 

1     +0        -3        -4    U^ 

+2        +4       +2 


1     +2 
+  2 


+  1 

4-8 


1     +4 

-1-2 
1     +6 


-2 


SOLUTION  OF  HIOHER  EQUATIONS  ill 

The  transformed  equation  is 

2/^  +  62/2  +  9^-2  =  0,  (2) 

in  which  y  =  x  —  2. 

Since  x  lies  between  2  and  3,  y  must  lie  between  0  and  1. 
Hence  y^  and  y^  are  smaller  than  y,  and  we  may  obtain  a 
rough  approximation  of  y  by  neglecting  the  first  two  terms 
of  equation  (2). 

I.e.  9y  =  2,  or  y  =  .2+ 

If  2/  =  .2,  we  find  f(y)  =  +  .048  ;  if  2/  =  0,  /(i/)  =  -  2. 

The  value  of  y  must  therefore  lie  between  0  and  .2,  or  y<  .2. 

Substituting  y  =  .1,  we  obtain  /(?/)  =  —  1.039. 

Hence  y  lies  between  .1  and  .2,  or  y  =  .1%  i.e.  a;  =  2.1'*'. 

Diminishing  the  roots  of  (2)  by  .1,  we  have 


1 

+  6.      - 
.1 

f   9. 
.61 

-2. 

.961 

1 

6.1 
.1 

9.61 
.62 

- 1.039 

1 

6.2 

•1 

10.23 

1        6.3 

The  second  transformed  equation  is 

23  ^  6  3  -^2  _^  iQ  23  2  _  1.039  =  0,  (3) 

in  which  z  =  y  —  .l,  i.e.  2;<.l. 

Consequently  we  obtain  an  approximate  value  of  z  by  neg- 
lecting z^  and  z^  in  equation  (3). 

10.23 2  =  1.039,  or  2=.l. 

But  as  this  value  is  too  large,  we  assume  z  =  .09. 


478 


ADVANCED  ALGEBRA 


Diminishing  the  roots  of  (3)  by  .09, 


1 

6.3 
.09 

10.23 
.5751 

- 1.039 
.972459 

1 

6.39 
.09 

10.8051 
.5832 

-   .066541 

1 

6.48 
.09 

11.3883 

I  .09 


1     6.57 
The  third  transformed  equation  is 

u^  +  6.57  ^2  + 11.3883  u  -  .066541  =  0, 
in  which  u  =  z  —  .09,  i.e.  w  <  .01. 


(4) 


Hence  we  obtain  an  approximate  value  of  u  from  the  equation 
11.3883  w  =  . 066541. 
w  =  .005+ 
Diminishing  the  roots  of  (4)  by  .005, 

.005 


1 

6.57 
-005 

11.3883 
.032875 

-  .066541 
.057105875 

1 

6.5/5 
.005 

11.421175 
.032900 

-  .009435125 

1 

6.580 
.005 

11.454075 

(5) 


1     6.585 

The  fourth  transformed  equation  is 

v'  +  6.585  v'  + 11.454075  v  -  „009435125  =  0, 

in  which  v  =  u  —  .005,  or  -y  <  .001. 

Hence  t^  and  v^  are  so  much  smaller  than  v,  that  usually  three 
significant  places  of  v  may  be  found  by  solving  the  equation 

11.454075  V  =  .009435125.  (6) 


SOLUTION  OF  HIGHER  EQUATIONS 

Therefore  v  =  .000823. 

Whence  a;  =  2.195823. 


479 


569.    The  work  is  usually  arranged  as  follows : 

2.195823 


1  0. 

-3. 

-4. 

2. 

4. 

2. 

1  2. 

1. 

-2. 

2. 

8. 

1  4. 

9. 

2. 

1    6. 


1    6.585 


Li 


.1 

.61 

.961 

1  6.1 
.1 

9.61 
.62 

-1.039 

1  6.2 
.1 

10.23 

1 

6.3 

.09 

10.23 

.5751 

- 1.039 

.972459 

1 

6.39 
.09 

10.8051 
.5832 

-  .066541 

1 

6.48 
.09 

11.3883 

.09 


1  6.57 

.005 

11.3883 

.032875 

-  .066541 

.057105875 

1  6.575 
.005 

11.421175 
.032900 

-  .009435125 

1  6.580 
.005 

11.454075 

I  .005 


^^.009435125^^^3^3 
11.454075 


480  ADVANCED  ALGEBRA 

570.  In  finding  the  approximate  value  of  the  second  figure 

of  the  root  {i.e.  y),  the  student  should  prove  the  location  of 
the  root  by  actual  substitution.  The  signs  of  two  functions, 
including  a  root,  should  be  opposite  (with  the  rare  exception 
of  two  nearly  equal  roots). 

571.  Any  figure  after  the  second  can  usually  be  found  by 
dividing  the  last  term  with  its  sign  changed,  by  the  coefficient 
of  X.  If  it  is  at  all  doubtful  which  of  two  successive  numbers 
should  be  assumed,  take  first  the  greater  one,  and  if  this  should 
be  too  large,  the  fact  can  be  recognized  by  the  following  two 
rules : 

(a)  In  any  transformed  equation  after  the  first,  the  signs  of 
the  last  two  terms  must  be  opposite.  If  they  should  be  the 
same,  the  value  assumed  for  the  root  is  too  large. 

(6)  Unless  the  equation  has  two  nearly  equal  roots,  the  signs 
of  the  last  terms  of  all  transformed  equations  must  be  the 
same. 

572.  After  the  equation  has  been  transformed  four  or  more 
times,  several  additional  decimals  can  be  found  by  division  of 
the  last  terms. 

In  equation  (5)  we  may  express  the  true  value  of  v  as  follows  : 
^.009435125     6.585 ^^.j.  ^3 
^~  11.451075         11.454075 

We  assumed  for  v  only  the  first  term  of  the  right  member,  hence  the 
error  is  equal  to  6.585  .^  +  .3 

11.454075 

A  simple  calculation  shows  that  this  error  for  v  =  .0008  is  less  than 
.000001,  an  error  which  cannot  affect  the  first  six  decimals  of  the  root. 
Therefore  the  three  decimals  obtained  by  division  are  correct. 

573.  To  avoid  decimals,  multiply  the  roots  of  the  first  trans- 
formed equation  by  10 ;  i.e.  multiply  the  second  coefficient  by 
10,  the  third  by  100;  etc. 


SOLUTION  OF  HIGHER  EQUATIONS 


481 


The  roots  of  the  second  transformed  equation  obtained  there- 
from, are  again  multiplied  by  10,  etc. 


574.   The  work  may  then  be  arranged  as  follows 


-3  -4 

4  2 


1 

2 

1 

-2 

2 

8 

1 

4 
2 

9 

1 

60 

900 

-2000 

1 

61 

961 

1 

61 

961 

-1039 

1 

62 

1 

62 
1 

1023 

1 

630 
9 

102300 
5751 

-  1039000 
972459 

1 

639 
9 

108051 
5832 

-   66541 

1 

648 
9 

113883 

1 

6570 
5 

11388300 
32875 

-  66541000 
57105875 

1 

6575 
5 

11421175 
32900 

-  9435125 

1 

6580 
5 

11454075 

1    6585 


2i 


9435125 
11454075 


2.195823 


LI 


[1 


=  .823. 


482 


ADVANCED  ALGEBRA 


575.  If  we  wish  to  determine  only  a  certain  number  of  deci- 
mals, the  work  may  be  further  contracted  by  omitting  all  figures 
which  do  not  influence  these  decimals. 


576.  Incommensurable  negative  roots  can  be  found  by  chang- 
ing the  signs  of  all  roots  by  the  transformation  of  §  531.  and 
determining  the  corresponding  positive  roots. 

Ex.  By  Horner's  method,  find  the  real  value  of  V  — 3  to 
three  places  of  decimals. 

Let  X  =  y^^. 


Then  x^  -\-  S  =0.    Since  the  required  root  is  negative,  Chans' 
of  the  roots,  i.e.  y^  —  S  =  0. 

10  0  -3  [1 

11  1 


'gr 


1 

1 
1 

1 

2 

-2 

1 

2 

3 

1 

1 

30 

300 

,  H  »'. 

■  4 

4 

13^ 

'  ,44 

1 

34 
4 

i52 

256 

-■■ 

38 
4 

588 

1 

520 

58800 

-  256000 

|4 

4 

2096 

+  243584 

1 

524 
4 

60896 
2112 

-  12416 

1     528         63008 
\^^  =  1+.     Hence  y  =  1.441+  and  x 


1.441+. 


EXERCISE  169 


In  the  following  equations,  compute  to  three  decimal  places 
the  root  which  lies  between  the  indicated  limit : 


SOLUTION  OF  HIGHER  EQUATIONS  488 

1.  a^— 15aj2  +  63a;  — 50  =  0,    root  between  1  and  2. 

2.  a.*^  —  12  a.-^  +  45  £c  —  53  =  0,    root  between  2  and  3. 
•  3.    ar^  —  12  a;^  +  45  a?  —  53  =  0,    root  between  3  and  4. 

4 .  ar^  — 12  a.-^  +  57  a;  —  94  =  0,    root  between  3  and  4. 

5.  ar^  —  12  a;  —  28  =  0,  root  between  4  and  5. 

6.  a^  -  15  aj2  4- 72  a; -109  =  0,  root  between  6  and  7. 

7 .  ar'  - 13  a;-  +  38  a;  + 17  =  0,    root  between  7  and  8. 

8.  ar^  — 12ar^4-35a;  — 73  =  0,    root  between  9  and  10. 

9.  ar^  +  2a.'2  +  3aj  — 52  =  0,        root  between  2  and  3. 

Find  to  three  decimal  places  the  real  root  of  the  following 
equations : 

10.  a;3_i2a;-132  =  0.  13.    a^  +  7a;-ll  =  0. 

11.  a^H-2a;-4  =  0.  14.   2a^  +  3a;  +  4  =  0. 

12.  a:3  +  3a;-2  =  0. 

Compute  the  roots  to  five  decimal  places : 

15.  a^  —  4  a^  4- 18  =  0,  root  between  3  and  4. 

16.  a;^  —  4  a;^  + 18  =  0,  root  bet  ween  ^2  and  3. 

17.  aj* 4- 8 a^ H- 16 aj  —  440  =  0,        root  between  —  4  and  —  5. 

18.  a;*-2a:3  —  9a^-f-10a;  +  7  =  0,  root  between  —  2  and  —  3. 

19.  a?*  — 17  =  0,  root  between  2  and  3. 

Find  by  Horner's  method  to  three  decimal  places  the 
value  of: 

20.  ^3.  21.    </7.  22.    V^-^. 

23.  The  number  of  cubic  feet  contained  in  a  cube  exceeds 
the  number  of  feet  in  all  its  edges  by  17.  Find  the  edge  of 
the  cube  to  three  decimal  places. 

24.  Find  two  numbers  differing  by  4  whose  product  is  equal 
to  1386  divided  by  their  sum. 


484  ADVANCED  ALGEBRA 

25.  The  number  of  cubic  feet  contained  in  a  cube  exceeds 
the  number  of  square  feet  in  its  surface  by  100.  Find  the 
edge  of  the  cube  to  three  decimal  places. 

GRAPHIC   SOLUTION  OF   CUBIC   EQUATIONS* 

577.  In  the  following  section,  a  graphic  method  for  solving 
cubic  equations  is  given,  which  is  somewhat  shorter  and  more 
convenient  than  the  usual  graphic  method.  It  consists  in  solv- 
ing equations  by  means  of  straight  lines  and  one  standard 
curve  which  is  identical  for  all  cubics. 

The  method  was  applied  to  quadratics  in  §  330,  and  it  can 
also  be  applied  to  equations  of  the  fourth  degree.f 

578.  Consider  the  cubic  aoc^ -{- bx -\-  c  ==  0,  (1) 
Let  y  =  Q^.  (2) 
Then                                    ay -\- bx -^  c  =  0.  (3) 

The  solution  of  the  system  (2),  (3)  for  x  produces  the 
required  roots. 

But  the  graph  of  (3)  is  a  straight  line,  while  the  graph  of  (2) 
is  identical  for  all  cubic  equations.  Hence  after  the  graph 
y  =  a^  (AOF  in  the  diagram)  has  been  constructed,  any  cubic 
may  be  solved  by  the  construction  of  a  straight  line. 

Ex.  1.    Solve  4  a^  -  39  a;  +  35  =  0.  (1) 

Let  y  =  x^  (2) 

Then  4?/-30aj  + 36  =  0.  (3) 

In  (3),  if  x  =  0,  then  y  =  -  8f,  and  if  x  =  4,  then  y  =  30^.  The  line 
joining  (0,  -  8f)  and  (4,  30J)  intersects  the  graph  of  (2)  in  P,  P',  and 
P'!.  By  measuring  the  abscissas  of  P,  P',  and  P",  we  find  a;  =  -3J,  or 
+  1,  or  2^. 

*  The  remaining  sections  of  the  book  are  not  absolutely  necessary  for 
the  Examinations  of  the  College  Entrance  Examination  Board. 

t  The  following  section  is  taken  from  a  paper  read  by  the  author  before 
the  American  Mathematical  Society  in  April,  1906. 


SOLUTION  OF  HIGHER  EQUATIONS 


486 


45 

Y 

/     ' 

"5    / 

"/ 

<* 

i 

/ 

N 

^^ 

1 

/ 

\ 

>> 

P' 

t 

\ 

N, 

/ 

t 

X 

V/r 

/ 

-4 

-3 

-2 

-1 

P 

/ 

/^ 

^ 

3 

* 

X 

/ 

t 

/ 

/ 

X 

/ 

/ 

^^1  ct 

/ 

/ 

/ 

/ 

}/ 

/ 

/ 

/ 

/ 

/ 

/ 

k^^ 

/ 

/ 

/A 

> 

/ 

• 

A(\ 

^ 

^ 

-45 

Y 

579.   In  the  equation  o!/  +  6*  -f-  c,  if  25  =  0,  then  y 


if 


y  s=  0,  then  aj  St  — .     Hence,  by  tajting  on  the  a>-axi8  the  point 

h 
—  -,  on  the  2/-axis  the  point  — ,  and  applying  a  straight  edge, 

h  a 

the  roots  of  the  equation  aa^  +  6a;  4-  c  ==  0  can  frequently  be 
determined  by  inspection.  If  the  two  points  thus  constructed 
on  the  axis  lie  very  closely  together,  the  drawing  is  likely  to 
be  inaccurate,  and  it  is  better  to  locate  one  or  both  points  out- 
side the  axis. 


486  ADVANCED  ALGEBRA 

Ex.  2.    Solve  af  +  6x-15  =  0.  (1) 

Let  y  =  x\  (2) 

Then  ?/  +  6x-15  =  0.  (3) 

Hence  the  distances  cut  off  by  (3)  on  the  x-  and  ?/-axes  are  respectively 
2\  and  15,  and  the  line  (3)  is  easily  constructed.  As  there  is  only  one 
point  of  intersection,  §,  the  equation  has  only  one  real  root,  viz,  1.7+. 

580.  The  curved  line  which  represents  the  equation  ?/  =  a;^  is 
a  cubic  parabola. 

The  curve  y  =  x^  used  in  §  330  is  a  parabola. 

581.  Solution  for  large  roots.  Since  a  linear  equation  is  repre- 
sented by  a  straight  line,  whether  the  abscissas  and  ordinates 
are  drawn  on  the  same  scale  or  different  scales,  we  can  use  the 
same  diagram  of  the  cubic  parabola  y  =  ^  for  the  finding  of 
large  and  small  roots. 

For,  in  the  diagram  we  can  assign  any  values  to  the  abscis- 
sas, provided  the  corresponding  ordinates  are  the  cubes  of  the 
abscissas. 

582.  Thus,  after  the  cubic  parabola  y  =  x^  has  been  drawn, 
we  may  multiply  the  numbers  on  the  ic-axis  by  any  convenient 
number,  e.g.  3,  and  the  values  of  the  ordinates  by  the  cubes  of 
the  number,  i.e.  27. 

Note.  Similarly  in  the  diagram  on  page  309,  the  value  of  the  abscissas 
may  be  multiplied  by  any  convenient  number,  e.g.  10,  provided  the  ordi- 
nates are  multiplied  by  the  square  of  that  number,  i.e.  100.  The  diagram 
may  then  be  used  for  roots  between  —  60  and  +  60. 

Ex.  3.    Solve  graphically  a^  -f  2  a;  -  320  =  0.  (1) 

The  superior  limit  is  v^320  or  7+,  and  there  can  be  no  negative  roots. 
Hence  multiply  the  values  of  the  abscissas  in  the  diagram  by  2  ;  then  the 
values  of  the  ordinates  have  to  be  multiplied  by  8.  (The  resulting  values 
are  given  in  parentheses.) 


SOLUTION  OF  HIGHER  EQUATIONS 


487 


Let 
Then 


y-¥2x 


y  =  x^. 
320  =  0,  or  1/  =  320  -  2  x. 


(2) 
(3) 


If  x  =  0,  y  =  320,  and  if  a;  =  8,  y  =  304. 

Joining  the  points  (0,  320)  and  (8,  304),  we  obtain  the  real  root  6.8-, 
while  the  other  roots  are  imaginary. 


-4P 

-35- 
-30 
-25 
-20 
-^1-5 
-10 

(360) 
f3«0> 

'-i-l-i 

1"=TI 

J 

1 

(280) 
(-240) 
(200) 
(160> 
(1-20) 
-(80) 

i 

> 

/ 

i 

1 

L 

// 

/ 

/ 

/ 

^w> 

r^ 

y 

1 

4 

0 

(])       (2)       (3)       (4)       ( 

)     (< 

'"  <P  "" 

Note.  The  student  should  draw  the  graph  of  y  =  a^  from  a;  =  —  3J  to 
a;  =  +  3 1  (or  from  —  4  to  +  4)  on  a  large  scale,  and  use  one  curve  for  the 
solution  of  a  number  of  equations.  The  table  on  page  288  will  be  found 
useful  for  the  construction,  as  it  gives  the  cubes  of  integers  and  fractions  ; 
thus  2.33  is  found  to  be  12.167. 


EXERCISE    170 

Find  graphically  the  real  roots  of  the  following  equations  : 

1.  ar^  +  4a;-16  =  0.  5.   a^-7a;  +  6  =  0. 

2.  ic3-5a;-12  =  0.  6.   4  a^-39a;-35  =  0. 

3.  a^_2a;  +  4  =  0.  7.   a^-5a;  +  20  =  0. 

4.  2a^-9x-^27  =  0.  8.   a^-5x-15  =  0. 


488  ADVANCED  ALGEBRA 

9.  r^-5x-5  =  0.  17.  a^4-10a;-13  =  0. 

10.  a;3-32a;-80  =  0.  18.  a^ -4.5  x -152  =  0. 

11.  2x'-5x-\-20  =  0.  19.  a^-60a;  +  180  =  0. 

12.  x^-^Sx-64:  =  0.  20.  a^-90a.'  +  340  =  0. 

13.  a^-10x-4:H  =  0.  21.  a;^  -  75  a;  -  250  =  0. 

14.  ar^-9a;-f  54  =  0.  22.  a;^^  -  100  a;  +  500  =  0. 

15.  a^-14a;  +  24  =  0.  23.  ar  + 120  a;  -  560  =  0. 

16.  aj«-30a;-18  =  0.  24.  ar' - 200 a;  +  1200  =  0. 

Transform  into  equations  without  a^,  and  solve  graphically : 

25.  a;3-6x'2  +  lla;-6  =  0. 

26.  x^-{-Sx^-4.x-\-l  =  0,^ 

27.  a^  +  9a^-7  =  0. 

Determine  graphically  the  character  of  the  roots  of  the  fol- 
lowing equations : 

28.  a;3_3^_5^0.  30.    a.'^-S  a;  +  5  =  0. 

29.  a^  +  3a;-5  =  0.  31.    x3  +  3a;  +  5  =  0. 


583.    Graphic  representation  of  a  cubic  function. 

Consider  the  equation 

a^-hpx  +  q=:0.  (1) 

Let  2/  =  ar^.  (2) 

Then  y+px  +  q=:0,  (3) 

or  y  =  —px—q. 

In  the  annexed  diagram,  let  COD  represent  the  cubic  parab- 
ola y=x^,  and  BE  the  straight  line  ?/  +  pa; + <7 = 0,  or  ?/  =  —px — q. 


SOLUTION  OF  UlGUEli  Ei^UATLONS  489 

Let  OA  or  x'  be  any  particular  value  of  a?, 
then  CA==x^\ 

and  BA  =  —px^  —  q. 

Hence  (75  =  CJl  -  5^  =  a;'^  +  px^  +  g. 

I.e.  f/ie  vcdue  of  the  furwtioji  x^-\-px-^q  for  any  particular 
value  x'  is  reptresented  by  that  part  of  the  corresponding  ordinate 
ivhich  is  intercepted  by  the  straight  line  y -\-px-\-q  =  0,  and  the 


' 

/j 

/c 

&' 

^'^tpx+Q 

/- 

t 

c 

V 

^^ 

>-p  x-(Z 

_ 

2 

0 

^ 

^ 

L 

^>x 

> 

^'' 

rx^ 

u 

>^ 

a 

^  ^ 

G 
/ 

/h 

<^ 

^ 

%^ 

k^ 

vV 

'm 

cw6ic  parabola  y  =  x^.  The  distance  is  measured  from  the 
straight  line,  and  is  taken  positive  if  it  extends  upward, 
negative  if  it  extends  downward. 

Thus  in  the  annexed  diagram,  f(x)  =  a^  —  ^^-x-\- 1,  and  we 
have/(-  2)  =  FG=^5,  /(-  l|)  =  iT/=  7,  fm)=KL  =  -  2,  etc. 

Ex.  1.   Find  the  greatest  value  of  the  function  a^  — 7a;H-6, 
for  a  negative  x.     (See  diagram  on  page  490.) 

Construct  J5,  the  locus  ofy  —  7a;  +  6  =  0-    Draw  CD  parallel  to  AB, 
touching  the  cubic  parabola  in  E,  then  FE,  or  14,  is  the  required  value. 

Ex.  2.    Which  values  of  x  will  make  the  function  x^  —  7x 
H-6equalto4,  t.e.         s(^-7x  +  6  =  4:? 


490 


ADVANCED  ALGEBRA 


On  any  ordinate,  from  the  straight  line  AB,  lay  ofE  4  units  upwards,  as 
FG.  Through  G  draw  HI  parallel  to  AB^  intersecting  the  cubic  parabola 
in  P,  P,'  and  P".  By  measuring  the  abscissas  of  P,  P',  and  P",  we  find 
K  =  —  2|,  or  ^,  or  2|. 


Note.  If  we  consider  the  distances  cut  off  from  SB  by  the  ordinates, 
as  abscissas,  e.g.  ST  =  1|,  then  the  cubic  parabola  represents  the  function 
jc^  +  px  +  g  in  so-called  "  oblique  coordinates." 

584.  To  construct  the  graph  of  ix^-j-px-^-q  in  the  usual 
manner  ("  rectangular  coordinates  "),  make  K'L'  =  KL,  M'N' 
=  MN,  O'B'  =  OB,  etc.  The  curve  L'JST'B'  is  the  required 
graph  of  a^+px  +  g. 

585.  The  value  of  the  function  ax^  -{-bx-^c  is  equal  to  a  times 
the  part  of  the  corresponding  ordinate  lohich  is  intercepted  by  the 
straight  line  ay -\-bx-\-c  =  0,  and  the  cubic  parabola  y  =  x^. 

The  proof  is  similar  to  that  of  §  585. 

Note.  This  method  of  constructing  graphically  the  value  of  a  function 
may  be  applied  in  a  similar  manner  to  quadratics. 


SOLUTION  OF  HIGHER  EQUATIONS  491 

EXERCISE   171 
Find  graphically : 

1.  The  value  of  a^+4  x-16,  if  x  equals  -3,  -2.5,  -2.1,  3.5. 

2.  The  value  of  arV4  x-8,  if  x  equals  -1.6,  -1.5,  2, 1.5. 

3.  The  value  of  ic^  -  6  a;  - 15,  if  a;  =  -  3,  -  2,  1.5,  3.5. 

4.  The  Y3\\ieoia^-5x-\-lS,iix  =  -S,  -5,  +3,  +7. 

5.  The  value  of  a;,  if  a^  —  5  a;  — 12  =  5. 

6.  The  value  of  a;,  if  a^- 5  aj- 12  =  -10. 

7.  The  value  of  a;,  if  a^- 5  a; -12  =  -40. 

8.  The  value  of  a;,  if  aj3- 5  a; -12  =  10. 

9.  The  smallest  value  of  a^— 5a;  — 12  for  a  positive  x. 

10.  The  greatest  value  of  a^  —  5  a;  +  10  for  a  negative  x. 

11.  Construct  the  graph  of  a;^  —  12  a;  —  30  =  0. 

12.  Construct  the  graph  of  a;^  —  8  =  0. 

Find: 

13.  The  value  of  2  a;^  +  9  a;  +  20  =  0,  if  a;  equals  3,  2.5,  - 1.5. 

14.  The  value  of    3  a;3  +  9a;-25  =  0,  if  x  equals  -3,  -5, 

-2. 

15.  The  smallest  value  of  3  a;^  —  9  a;  —  25,  for  a  positive  x. 

16.  The  character  of  the  roots  of  the  equation  a^  —  7a;  —  5  =  0. 

17.  Locate  the  roots  of  the  equation  4  ar'  —  80  a;  +  570  =  0. 

18.  For  which  values  of  a;  is  4  a;^  —  80  a;  +  570  =  0  positive  ? 

19.  Construct  the  locus  of  a;^  — 14  .r  +  20  =  0. 

20.  Construct  the  locus  of  a;^  — 15  a;  —  20  =  0. 


492  ADVANCED  ALGEBRA 

GENERAL   SOLUTION  OF   CUBIC   EQUATIONS 

586.  The  methods  used  in  the  preceding  chapters  relate  to 
numerical  equations  only,  but  they  cannot  be  used  to  find  gen- 
eral  solutions,  i.e.  the  values  of  the  roots  of  literal  equations  in 
terms  of  the  coefficients. 

The  only  general  solution  given  thus  far  is  the  solution  of 
the  quadratic  ax^  -\-  bx-\-c  =  Oj  (1) 


viz.  ^^-6±Vy-4ac.  (2) 

This  solution  is  general  because  any  quadratic  may  be  con- 
sidered a  special  case  of  (1),  and  the  roots  of  any  equation 
may  be  found  by  substituting  in  (2)  the  particular  values  of 
the  coefficients  in  place  of  a,  b,  and  c. 

587.  It  is  possible  to  give  general  solutions  of  equations  of 
the  third  and  fourth  degree,  but  it  was  first  proved  by  Abel* 
that  it  is  impossible  to  obtain  general  algebraic  solutions  of 
equations  higher  than  the  fourth  degree. 

588.  The  cube  roots  of  unity.  To  find  the  cube  roots  of 
unity,  we  have  to  solve  the  equation 

a^  =  l,  oric3-l  =  0.  (1) 

Factoring,       (a;  —  l)(ic*  -f  a;  +  1)  =  0. 

Hence  a?  —  1  =  0, 

or  af  +  x-\-l=0.  (2) 

Whence  x  =  1,  or  x  =  ~  ^  (1  ±  i  V3). 

Let  _  j(l-|.tV3)  =  a).t 

Then  0)2  =  J  (1  + 1  Vsy  =  _  1  (1  -  /  V3). 

Hence  the  three  cube  roots  of  unity  are  1,  w,  w^ 

*  For  an  elementary  representation  of  this  proof,  see  Peterson,  Theorie 
der  Algebraischen  Gleichungen,  p.  113.  t  The  Greek  letter  Omega. 


SOLUTION  OF  HIGHER  EQUATIONS  493 

589.  If  one  cube  root  of  any  quantity  is  r,  the  two  others 
are  rw  and  rw^. 

For  ^a  =  ^^Tl  =  -^a  •  ^1  =  r^I, 

i.e.  r,  r<i),  rw^,  are  the  three  cube  roots  of  a. 

590.  In  the  annexed  diagram,  the  lines  OAj  OA',  OA"  rep- 
resent the  three  cube  roots  of  unity 
graphically  in  the  plane  of  complex 
numbers.  The  length  of  each  line 
is  1,  and  the  angles  included  by 
them  are  120°. 

If  OB  represents  the  real  value 
of  ^a,  then  OB'  and  OB"  repre- 
sent the  other  two  cube  roots  of  a. 

591.  Cardan's  solution  of  the  cubic  equation.     By  §  565,  any 

cubic  can  be  reduced  to  the  form 

»^+p»  +  g  =  0.  (1) 

Suppose  y  =  m  +  7if 

then  2/^  =  m^  4-  3  mn  (m-\-n)  ■}-  n^. 

Substituting  for  m  +  n  its  value,  and  transposing, 

f-Smny-~(m^-\-n^)  =  0.  (2) 

If  we  make  the  coefficients  of  equation  (2)  equal  to  the 
coefficients  of  (1),  the  two  equations  become  identical,  and 
y  =  x. 

Or,  if  3mn  =  -p,  (3) 

and  m^  +  7i^  =  — g,  (4) 

then  a;  =  m  +  n.  (5) 

But  m  and  n  are  easily  found  from  (3)  and  (4). 

From  (3)  4mV  =  -i|'.  (6) 


494  ADVANCED   ALGEBRA 


Squaring  (4)  and  subtracting  (6), 

(7) 

Hence           m^ 
But                m^ 

+  n'  =  -q. 

(8) 

Therefore 

»^=-i4V^^+f.- 

' 

-=^/-|WM?•      . 

Similarly 

3;                    .              .            3; 

or 


Therefore  «;  =  «+«=  V-|  +  Vj  +  ^  +  V" |- V|'  +  Jr 

This  result  is  known  as  Cardan^ s  formula,  although  Cardan 
did  not  discover  it.  He  published  it  in  1545  as  his  own  work, 
after  having  obtained  it  under  the  pledge  of  secrecy  from  Tar- 
taglia. 

592.  Since  m  and  n  are  cube  roots,  they  have  three  values, 
and  if  r  and  r'  represent  respectively  one  root,  then 

m  =  r,  or  rw,  or  rw^, 

71  =  r'  or  r'co,  or  r'w^. 

If  every  value  of  m  could  be  combined  with  every  value  of  w, 
there  would  be  nine  values  of  x ;  but  since  3  mn  =  —  p,  the  prod- 
uct of  m  and  the  corresponding  value  of  w  must  be  real.  Hence 
each  value  of  m  can  be  combined  with  only  one  value  of  n,  thus 
producing  the  roots 

x  =  r-\-r'j 

or  x  =  r<D  -\-  rw'^f 

or  x  =  r<i)^  -f  r'io. 


SOLUTION  OF  HIGHER  EQUATIONS  495 

593.  If  all  three  roots  are  real  and  unequal,  Cardan's 
formula  gives  the  answers  as  the  sum  of  two  cube  roots  of 
complex  numbers.  As  there  is  no  general  algebraic  method 
for  reducing  this  answer  to  a  real  form,  this  case  is  known 
as  the  irreducible  case.  Horner's  method  should  then  be 
employed. 

Ex.  1.    Solve  the  equation 

a^  -  6  a;  -  9  =  0.  (1) 

The  shortest  way  is  to  substitute  the  values  of  p  and  q  in  Cardan's 
formula.  Since  this  formula,  however,  is  difficult  to  remember,  we  pro- 
ceed as  follows : 

Let  y3  =  (wi  +  nY  =  wi'  +  3  win(m  +  w)  +  n*. 

Or  y3  -  3  mny  -  (m^  +  n^)  =  0.  (2) 

Equating  the  coefficients  of  (I)  and  (2), 

3  mn  =  6,  (3) 

m»  +  71^  =  9.  (4) 

From  (3)  4  m^n^  =  32.  ,  (5) 

Hence  m»  -  2  m^n^  +  n«  =  49.  (6) 

Or  m*-n^  =  7.»  (7) 

m»  =  8,  n8  =  1. 

Whence  m  =  2,  2  w,  2  w^. 

n  =  1,  w,  w2. 

Therefore  x  =  3,  2 « +  <^^  2  o;2  +  w.  (8) 

Substituting  the  value  of  w,  x  =  3,  or  -  i(3  ±  iVS).  (9) 

♦  The  value  —7  will  produce  the  same  value  of  a;  as  +7. 


496  ADVANCED  ALGEBRA 


EXERCISE  172 


Solve  by  Cardan's  method : 

1.  a;3_3^_2==o.  8.  a^-72a;-280  =  0. 

2.  a^-9a;  +  28  =  0.  9.  a^-6a;  +  9  =  0. 

3.  a^-18a;-35  =  0.  10.  a^  +  9ic-26  =  0. 

4.  a;3-36a;-91  =  0.  11.  a^-9x-2S  =  0. 

5.  a:^-3x-\-2  =  0.  12.  or^ -  72 a;  +  280  =  0. 

6.  x^-27x-54.  =  0.  13.  ar^-6a;2-12a;4- 112  =  0. 

7.  a;^  +  9x4-26  =  0.  14.  a^  + 5a^  + 8a;4-6  =  0. 

SOLUTION  OF  BIQUADRATICS 

594.  A  biquadratic  is  an  equation  of  the  fourth  degree. 

595.  Descartes'  solution  of  the  biquadratic. 

Let      x^-\-px^-{-qx  +  r  =  0.  (1) 

Assume  x'^ -\-px^ -Tqx-\-r=(p(?-\-lx-\-m){x^—lx-\- n)  (2) 

—  x'^-\-{—l'^-{-m-^n)x'^—l{m—n)x-\-mn. 

Equating  the  coefficients,  we  have 

—  l^-^m-\-n  =p,  ~l{m  —  n)^q,  mn  =  r.  (3) 

Considering  that  (m  +  ^)^  —  (m  —  n)^  —  4  mn  =  0,  it  is  easy 
to  eliminate  m  and  n.     We  obtain 

l^^2pl'-{-(y-4.r)l'-q'  =  0.  (4) 

Equation  (4)  is  a  cubic  in  P,  which  has  always  one  real 
positive  root.  When  I  is  known,  the  values  of  m  and  n  are 
determined  from  equation  •  (3).  The  solutions  of  the  two 
equations  a?  +  lx  +  m  =  0 

and  x^  —  lx-]-n  =  0 

produce  the  required  roots. 


SOLUTION  OF  HIGHER  EQUATIONS  497 

Ex.     Solve  a;^-6a^-H8a;-3  =  0. 
As  ;?  =  -  6,  q  =  S,  r  =  —  3,  Ms  found  from  the  cubic 
Z5  _  12  Z4  +  48  Z^  -  64  =  0. 

A  real  root  of  this  equation  is  l'^  =  4,  hence  I  =±2.    Taking  Z  =  2,  we 
find  from  equation  (3)  m  =  —  3,  n  =  1,  and  the  two  quadratic  equations 

^''®'  x^  +  2x-Z  =  0 

and  a;'^  -  2  X  +  1  =  0. 

Hence  the  required  roots  are  1,  1,  1,  and  —  3. 

EXERCISE  178 
Solve  by  Descartes'  method : 

1.  x^-x^-\-4:X-4:  =  0.  5.    x^-5a^-10x-6  =  0. 

2.  x*-4:X^  +  4:X-l  =  0.  6.    x*-4.af -{-12 X -9  =  0. 

3.  a^-6sc^-{-Sx-{-2  =  0.  7.   x*- 25 a^-\- 60x^36  =  0. 

4.  aj^- Tar' -12 a; +  18  =  0.         8.    a;^-60a:^+40a;-}-396=0. 

RECIPROCAL  EQUATIONS 

596.  An  equation  is  called  a  reciprocal  equation  if  the  recip- 
rocal of  any  root  is  again  a  root. 

In  a  reciprocal  equation  of  odd  degree:  one  root  must  be  its  own 
reciprocal,  hence  one  root  must  be  either  4- 1  or  —  1. 

597.  Let    f(x)  =  a^^-\-aiaf-'-\-'"-{-an-iX  +  a^  =  0  (1) 
be  a  reciprocal  equation. 

Substituting  -  for  a;, 

X 


«ol^T+a/r.Y  V-4- 


«n-ig)+««  =  0,  (2) 


.XJ  \x. 

or  «naJ"  +  a^-ia^-^  H |-aiaj  +  ao=0.  (3) 

2k 


498  ADVANCED  ALGEBRA 

Equation  (3)  lias  the  same  roots  as  equation  (1).  Hence  the 
corresponding  coefficients  are  equal  or  proportional. 

I.e.  ^  =  ^,  01-  ao'  =  al 

Therefore  a„  =  ±  Qq. 

Similarly  it  follows  that  a„_i  =  ±  «!,  etc. 

598.  Either  the  coefficieyits  of  terms  equally  distant  from  /he 
first  and  last  terms  are  equal  or  their  absolute  values  are  equal 
and  their  signs  opposite. 

Thus  2a;*-5x3  +  2x2~5x  +  2  =  0, 

3x5  -  4x*  +  2x3  +  2x2  -  4x  +  3  =  0, 

x5  +  x*-x-l  =  0 
are  reciprocal  equations. 

599.  A  reciprocal  equation  of  even  degree  having  the  signs 
of  the  first  and  last  terms  equal  is  called  a  standard  reciprocal 
equation. 

3x*-5x3  +  7x2-5x  +  3  =  0isa  standard  reciprocal  equation. 

600.  Any  reciprocal  equation  can  he  reduced  to  a  standard 
reciprocal  equation  by  removing  the  factors  x  —  1  or  x  +  1,  or 
both. 

Any  reciprocal  equation  of  odd  degree  has  the  roots  +  1  or  —  1  (§  599), 
hence  by  removing  the  factor  (x  —  1)  or  (x  +  1)  it  can  be  reduced  to  even 
degree. 

A  reciprocal  equation  of  even  degree,  v\rith  the  signs  of  the  first  and  last 
terms  opposite,  can  have  no  middle  term.  By  grouping  terms  it  can  easily 
be  shown  that  x^  —  1  is  a  factor  of  the  left  member.  By  removing  the 
factor  x^  —  1,  the  equation  is  reduced  to  a  standard  equation. 

601.  A  standard  reciprocal  equation  can  be  reduced  to  an 
equation  of  half  its  dimension. 

Let  2a;«-3a.'5H-5a;^-7a^  +  6iB2-3aj  +  2  =  0.  (1) 


6^0:2 +  l)-36(a:  +  l)  + 62  =  0. 


Let  X  +.-  =  y,  then  a;^  +  J-  =  y2  _  2. 

X  x^ 

Substituting,  6  y2  _  12  -  36  y  +  62  =  0. 


(2) 


SOLUTION  OF  HIGHER   EQUATIONS  499 

Dividing  by  a^, 

Grouping  terms, 

2(.3  +  i)-3(^  +  i)+5(.  +  l)-7  =  0. 

Let  y  =  «  +  i,then/  =  a^  +  2  +  l. 

Hence         a^ -|- i  =  ^^  _  2. 

=  2/(2/^-3)  =2/«-3z/. 
Substituting  in  (2) 

2(f-Sy)-3(f-2)+5y-7  =  0, 
Simplifying,  2^^ -3f -y-l  =  0, 

Ex.  1.    Solve  6ar'-29a;*  +  27ar»  +  27a^-29a;H-6  =  6. 

Evidently  a;  =  —  1  is  a  root. 
Dividing  by  x  +  1, 

6  X*  -  35x8  +  62x2  _  36x  +  6  =  0. 
Dividing  by  x^  and  grouping, 


600  ADVANCED  ALGEBRA 

Or  6  2/2  -  35  ?/  +  50  =  0. 


y  =  35dzVl225-1200^3^^^g^^ 

Hence  a;  +  1  =  3i,  or  a;  +  -  =  2^. 

a:  X 

Solving,  we  find  the  roots  1,  3,  |,  2,  ^. 


EXERCISE  174 
Solve: 

1.  Sx*-54:a^-{-101a^-54.x-}-S  =  0. 

2.  10ic*-77ar^4-150a^-77aj  +  10  =  0. 

3.  5a^-2a;2  +  2x-6  =  0. 

4.  a;^  +  5aj3  — 5a;  — 1  =  0. 

5.  a^-6x^-\-6x-l  =  0. 

6.  2a!4-9a^  +  14a;2-9a;  +  2  =  0. 

7.  4a;^-25a:^  +  42a;2_25aj  +  4  =  0. 

8.  6i»*-35«3  +  62a^-35x  +  6  =  0. 

9.  12aj4-91a^  +  194aj2_9i^_^12  =  0. 
10.  5x'-36a^-^62x^-S6x  +  5  =  0, 

11.  Find  five  values  of  VT. 

12.  Find  five  values  of  ^— 1. 


APPENDIX 

I.  MULTIPLICATION   BY  DETACHED  COEFFICIENTS 

1.  When  the  literal  part  of  the  product  of  two  polynomials 
can  be  obtained  by  inspection,  the  work  of  multiplication  may 
be  simplified  by  omitting  all  literal  factors. 

Ex.1.   Multiply  4 ic2  +  6 a:  +  2  by  2a;2^5a;-L  • 

ORDINARY    SOLUTION 

2x'-5x-l 
8a;*  +  12aj34.   4a^ 

-20a^-30a^-10x 

-   4a^-   6x-2 


8(B*-   Sa^-S0ay'-16x-2. 

DETACHED     COEFFICIENTS 

4       +6       +2 
2       -   5       -   1 


8       +12       +4 

_20       -30  -10 

-4  -   6      -2 

8       -8       -30  -16       -2. 

Hence  the  product  equals : 

Sx'-Sa^-S0x'-16x-2. 

2.  An  example  containing  an  irregular  sequence  of  expo- 
nents may  be  made  regular  by  the  insertion  of  zero  coefficients, 
e.g.  a:^-\-2x  +  l  =  a:^-\-0ar  +  2x  +  l. 

601 


502  ADVANCED  ALGEBRA 

Ex.  2.    Multiply  x^-^x^-7  by  2  a^  + 1. 

1  4-1      +0      -   7 
2+0+1 

2  +2       +0       -  14 

+1+1+0       -7 

2       +2       +1       -13       +0       -7. 

The  highest  exponent  of  the  product  is  obviously  5,  hence  the  product 
equals  2  x^  +  2  x*  +  a;^  -  13  ic^  -  7. 

EXAMPLES 

Perform  the  following  multiplications  by  detached  coeffi- 
cients : 

1.  (ic2  +  2a;  +  5)(a^-3ic4-l). 

2.  (x^-3x-7)(x'^-{-Sx-2). 

.3.  (2aj3-5a^  +  2a;  +  l)(a;2  +  3aj  +  l). 

4.  (a^  4-  3  a^ft  +  5  ab'  -  ?>=^) (a^  -  2  a&  +  b'^. 

5.  (a^  +  2ic-5)(a^-5a;-hl). 

6.  (6a2-2a64-5&2)(6a2  +  2a&-562). 

7.  (m%^4-6m^n^  +  9)(mW  +  5mn-t-l). 

3.   The  method  of  detached  coefficients  can  be  applied  to 
addition,  division,  extracting  of  square  roots,  etc. 


11.  ADDITIONAL  CASES  IN  FACTORING 

A.   Trinomials  of  the  type  p^x*  +  qx^^  +  ry*. 

The  trinomial  p  V  4-  qx^y^  4-  r^y*  can  be  factored  if  ±2pr  -  -  q 
is  a  perfect  square. 

Ex.  1 .   Factor  9  a;*  + 15  x^y^  + 16  2/*. 

9x*  +  15xV  +  16y*  =  9a;*  +  24a;V+  16!/*-9a;2ya 

=  (3x2  +  4y2)2_(3a;y)2 


APPENDIX  503 

Ex.2.   Factor   16 a;^ - 56 a^y^ ^_ 25^^. 

16  x*  -  5t)  x2y2  +  25  ?/»  =  10  X*  -  40  xV  +  25  y*  -  16  x^y 
=  (4  x2  -  6  2/2)2  _  (4  a-yyi 
=  (4  x2  +  4  xy  -  5 1/2)(4  ic2  _  4a;2/  -  6  y'^). 

EXAMPLES 
Factor : 

1.  a^  +  a?  +  l.  6.  ic*  +  2/*-14jc2/j 

2.  a^  +  14a2  +  81.  7.  9 aj*  + 11  a^/ +  4 2^. 

3.  64  a*  + 12  a^ -f  1.  8.  ^  a^ +  3a%''-^4.h\ 

4.  9«^-4«y4-42/*.  9.  a^-18a262  +  Z;^ 

5.  16a;*-25a;y4-92/*.  10.  4m*-28  mV  +  9  7i*. 

B.   Binomials  of  the  type  a"  ±  b".* 

1.  Binomials  of  the  form  a"  —  6"  can  always  he  factored. 
By  actual  division : 

^^iJ^  =  a^  4-  a^h  +  a^ft^  +  aW  +  6*. 
a  — 6 

Hence  a'-h'^{a-  b)  {a*  +  a»6  4-  «'&'  4-  «&'  4-  b*). 

Similarly,     a«  -  6«  =  (a  -  6)  (a^  +  a'b  4-  a^i'  4-  a%^  +  ab'  +  b'). 

Or  in  general, 

a**  -  6«  =  (a  -  ft)  (a'*-*  4-  a"~^2>  4-  a""^?>'''  H h  &""^)- 

2.  To  obtain  prime  factors,  it  is  better  to  consider  a^  —  6®  the 
difference  of  two  squares  than  the  difference  of  two  6th  powers. 

Thus,  a«  -  6«  =  (a«  4-  6')  (a'  -  b^ 

=  (a  4-  b)(a^-ab  4-  6'0 (a  -  6)(a2  4-a6  +  b^). 
Similarly,  a^°  —  6^**  should  be  considered  the  difference  of  two 
6th  powers. 

a'«  -  6^0  =  (a'  -  b^(a^  4-  «'&'  4-  a^6^  +  a^^'  +  b^ 
=  (a  +  6)(a-6)(a«..-). 

•  See  Chapter  XVI. 


504  ADVANCED  ALGEBRA 

3.  Binomials  of  the  form  a"  +  6"  can  be  factored  if  n  is  odd. 
By  actual  division : 

^!l±^  =  a*  _.  a^b  +  a'b^  -  ab^  +  b\ 
a  +  6 

Hence  a' -h  b' =  (a -{-  b)  (a'  -  a'b  +  a'b'  -  aW  +  6*). 

Similarly, 

V  +  57  ^  (a  +  6)(a«  -  a^6  +  a'W  -  a'b^  -f-  a^b'  -  ab'  +  &^). 

Or  in  general,  if  n  is  odd, 

^n  _|_  5«  ^  (^^  _|_  5)  (^n-l  ^  ^n-2^^^n-3^2 ^  ^n-l^j^ 

4.  TJie  sum  of  two  even  powers,  as  a'*  +  W,  a^  +  b^,  etc.,  is  not 
exactly  divisible  by  a  -\-  b  or  a  —  b. 

Frequently  such  expressions  can  be  considered  the  sum  of 
two  odd  powers. 

a^  +  b^=  (ay  +  (by  =  (a^  +  b^)  (a'  -  a%''  +  b"). 

^10  ^  JIO  ^  (^2y  _,.  Q^2y  ^  (^^2  _^  ^2^)  (^^8  _  ^6^2  _|.  ^454  _  ^256  ^  58), 
EXERCISE 

Divide : 

1.  ocl^  —  fhj  x  —  y.  5.  a^^  +  b^  hj  a'^  +  b\ 

2.  0^  +  1  by  a;  4- 1.  6.  1  +  w^  by  1  H-m". 

3.  m'  —  n'  hj  m  —  n.  7.  a^  —  6^  by  a^  —  61 

4.  a*-16bya-2.  8.  a^-b^hy  a-b. 

Factor : 

9.  m^-~n\  15.  32H-wV.  21.  a'^  +  l. 

10.  a^-b\  16.  m^"-l.  22.  a^  +  64. 

11.  c^  +  32.  17.  a«6«-c^  23.  a^^-fl. 

12.  «^~1.  18.  a^-b'^  24.  a^o+ftVo. 

13.  a^  +  1.  19.  a»-729.  25.  a^-/. 
,    14.  a^b^  +  c^,  20.  a^^  — a^. 


APPENDIX  505 

m.    HIGHEST  COMMON  FACTOR  AND  LOWEST 
COMMON  MULTIPLE 

1.  The  Highest  Common  Factor  of  two  expressions  which  cannot 
be  factored  by  inspection  may  be  found  by  a  method  analogous 
to  the  one  used  in  arithmetic  for  the  finding  of  the  greatest 
common  divisor  of  two  numbers.     (Euclidean  Method.) 

Suppose  it  is  required  to  find  the  H.  C.  F.  of  x*  —  5qi^  + 
4.x'^-^10x-12  anda^-3a;2_3a;  +  9. 

By  dividing  the  first  expression  by  the  second,  we  obtain 
the  quotient  x  —  2  and  the  remainder  x^—5x-\-6.     Hence 

(x'-5a^-\-4.a^A-10x-12)  =  (a^-^3x'-3x  +  9)(X'-2) 

-\-(x'-5x-\-6)  (1) 

From  (1)  follows  that  any  factor  contained  in  the  divisor 
(a:^  —  3x^  —  3x-\-9)  and  the  remainder  (x^  —  5x-\-6)  is  a  factor 
of  the  entire  right  member,  and  therefore  a  factor  of  the  left 
member.  Hence  a  factor  common  to  divisor  and  remainder  is 
also  a  common  factor  of  divisor  and  dividend. 

Similarly  it  follows  that  a  factor  common  to  dividend  and 
divisor  is  also  a  common  factor  of  divisor  and  remainder. 

Hence  the  required  H.  C.  F.  of  dividend  and  divisor  is  also  the 
H.  C  .F.  of  divisor  and  remainder. 

We  have  therefore  to  find  the  H.  C.  F.  of 

ar'-Sic^-Sic  +  O  and  a^-5a;  + 6. 

This  can  be  done  by  determining  whether  or  not  the  factors 
of  ic^  —  5  a;  —  6  are  also  factors  of  ar^  —  3  ar^  —  3  x  +  9. 
We  may,  however,  divide  again. 


a?-3x'-3x  +  <d 

x'-hx  +  Q 

a^_5iB2  4.6a; 

x  +  2 

2a^_9a;  +  9 
2ar^_10a;-+-12 

Kem.  =  X  —  3 

506 


ADVANCED  ALGEBRA 


It  follows  in  precisely  the  same  manner  as  before  that 
the  required  H.  C.  F.  must  be  the  H.  C.  F.  of  the  new  divisor 
x^  —  5  x-^6  and  the  new  remainder  x  —  S. 

Dividing  again,  we  obtain  no  remainder,  and  therefore  x  —  3 
is  H.  C.  F.  of  a;2  _  5  ^  _^  g  and  a;  -  3. 

Hence  the  required  H.  C.  F.  =  ic  —  3. 

The  work  of  the  last  example  may  be  arranged  as  follows : 


Quotients 

x'-5a^  +  4.x'  +  10x-12 

a^- 

-Sx'- 

3a;+   9 

a;-2 

X*  -Sa^-3x^-\-   9x 

a^- 

-5a^  + 

^x 

-2a^  +  7a^+       a; -12 

2x'- 

9a;+    9 

a;  +  2 

-2ar^  +  6a^+    6x-lS 

2^- 

10a;  +  12 

First  remainder, 

a^-   5x-^   6 

a;-3 

x-2 

a^-   3a; 

-   2x-^   6 

-    2a;+    6 

H.C.F.  =  a;-3 

2.  Before  dividing,  all  monomial  factors  of  the  two  given 
expressions  should  be  removed  and  their  H.  C.  F.  be  determined 
separately. 

All  monomial  factors  of  any  of  the  successive  divisors  should 
be  removed,  as  they  do  not  affect  the  answer. 

If  the  first  term  of  a  dividend  is  not  exactly  divisible  by  the 
first  term  of  the  divisor,  multiply  the  dividend  by  such  a 
number  as  will  make  the  term  divisible.  E.g.  the  first  term 
of  3  a;^  4-  5  a;  +  2  is  not  exactly  divisible  by  2  a;  +  3,  hence 
multiply  3x^  +  5  a; +  2  by  2,  and  divide  6a;2  +  10a;  +  4  by 
2  a; +  3. 

Since  the  divisors  do  not  contain  any  monomial  factors, 
the  introduction  of  such  a  factor  into  the  dividend  does  not 
produce  a  common  factor,  and  hence  does  not  affect  the 
result. 


APPENDIX 


507 


Ex.2.    Find  the  H.  C.  F.  of  ^x^y -^  x^f +  Qxy^  and  Q7?f 

+  24a;y-30  2/*. 

Quotients 

3a:y|3arty-  9a;V+6a;y* 
x3_  3xy2  +2  2/8 


6  3/2|6xVH24xV-30y5 
a;3+  4x-2y  -   5 1/8 
«»-  3x2/2  +21/8 

1/14x22/  +  3x2/2- 

-72/8 

4x2    +  3x2/  ■ 
4x2    _  4x2, 

-72/2 

7X2/- 
7X2/- 

-7  2/2 
-7y2 

H.  C.  F.  =  3  j/(x  -  2/). 

4x8-12x2/2  +8  2/^ 
4x8+   3x2?/  -7x2/2 


-  3x2?/  -5x2/2+  82/8 

-  4 


12x22/ +  20x2/2 -32  2/8 
12x22/+  9xy2-2l2/8 

ll2/2|llx2/2-n2/8 
.    x-y 


x-f-3y 


4x-72/ 


Explanation.  The  monomial  factors  of  the  given  expressions  are  6  y^ 
and  3  xy  ;  their  H.  C,  F. ,  3  2/,  is  reserved  as  a  factor  of  the  answer.  Re- 
move from  the  second  divisor  (4  x^y  +  3  x2/2  —  7  2/8)  the  simple  factor  j/, 
and  multiply  the  second  dividend  (x8  —  3  xy^  -\-  2  y^)  by  4.  Similarly 
—  3  x2y  -  5  X2/2  +  82/8  is  multiplied  by  —  4,  and  the  factor  11  y^  is  re- 
moved from  the  last  divisor  11  X2/2  —  11  y*. 


EXERCISE 
Find  the  H.  C.  F.  of  the  following  expressions : 

1.  «3_2a^4.4a;  +  7  and  :x?-4:a? ^10 x-7. 

2.  aj»  +  3«*-3a;-5  and  ar' - 3 ar^ -f- 2 a;  +  6. 

3.  Sar'  +  a^  —  a;  +  l  and  15 a^  —  ar'  —  a; -f 3. 

4.  a;*4-a^  +  l  and  a;* -fa;. 

5.  a;*  +  a52-f  1  ^nd  ^y  —  y. 

6.  2a;3-5ar*-6a;  +  9  and  3ar'-2a:*  +  a;-2. 

7.  6a:3  4-13ar^  +  15aj-25  and  2a:«-f4a;2^4a._10. 

8.  ar'-4a;'-f  2a;  +  l  and  ar'-2ar'-f3a;-2. 

9.  a;^-5a:'4-5ar'  +  5a;-6  and  ar'-3«^-6x-f  8. 

10.   a*  +  4a«-9a2-16a-f  20  and  a»- 2a»-23a-f  60. 


50g  ADVANCED  ALGEBRA 

11.  a^-16ar''  +  86«2-176a;  +  10^  and  x^  -  19  x" -\- 12S  x^ 

-356  a; +  336. 

12.  »^-5ar'  +  5a^  +  5a5-6  and  a;^  +  2  ar' -  13  a;^  - 14  a;  +  24. 

13.  a;^  —  5  ax-^  +  5  a^af  +  5  a^x  ~6  a^  and  a?^  -f  4  aa;^  +  a^x  —  6  a^ 

14.  5a.'3-9a;2_5^_|.9  and  2a;^-a^-2a;-f-l. 

15.  2a^-2a52+6aj-6  and  2a?*-6a.-3  +  10aj2-6aj. 

16.  7  01^  —  4:  x^y  —  2  xy^  —  y^  and  3a^  —  Sx^y  -\-  xy^  —  y^. 

17.  a^-\-Sa'b-5alr'-\-b^  and  4  a^  +  2  a^ft  -  8  a*^62  _^  2  a^^^s^ 

18.  5a*  +  2Sa^b  +  23ab^  +  5b*  and  3  a^  +  14a»ft +  9a6^  +  26*. 

19.  4aj*  +  7a^  +  5aj2-a;-3  and  4aj*  +  5.7^  +  3a;2-2. 

20.  2iC*-8a;^  +  a;2  +  llaj  +  3  and  2x^-4a:3_^^2_|_^_3^ 

21.  a^_af^_3aj2-19a;-10  and  a;^- a^- 2  a;=- 17a;-5. 

22.  a;^  +  5ar^  +  lla^  +  13a;  +  6  and  a^-a^-3a;-9. 

Eeduce  to  lowest  terms : 
„,     2a^-a^2_-i^g^_l_;L5  a^  + 3  x^  -  4  a;-12 


a;*  +  3x»-aj-3  a;^  +  5a^  +  5a;2-5a;-6 


3.  The  Lowest  Common  Multiple  of  two  expressions  which 
cannot  be  factored  by  inspection  can  be  found  in  the  following 
manner : 

Let  A  and  B  denote  any  two  expressions  whose  H.  C.  F.  is  c, 

and  let  —  =  a  and  —  =  6. 
c  c 

Then  A  =  a  -  c,  B  =  b  >  c. 

Hence  L.  C.  M.  =  a  •  c  •  6 

c 


APPENDIX  509 

To  find  the  H.  C.  F.  of  the  two  expressions,  divide  one  of  the 
expressions  by  their  H.CF.  and  multiply  the  quotient  by  tlie 
other  expression. 

Thus,  to  find  the  L.  C.  M.  of  «»  -  2  x2  +  a;  +  4  and  x^  -  3  x2  +  2  x  +  6, 
determine  their  H.  C.  F.  (x  +  1). 

Dividing  x^  —  2  x^  +  x  +  4  by  x  +  1,  we  obtain  .x^  -  3  x  +  4. 
Hence,  the  L.  C.  M.  =  (x2  -  3 x  +  4)(x3  -  3 x2  -f  2 x  +  6). 

EXAMPLES 
Find  the  L.  C.  M.  of  the  following  expressions : 

1.  a^  +  7ay^-^nx-j-2  ?i,nd  a^-\-8s(^-\-16x-{-S, 

2.  a^  +  a^-a  +  lo  and  a^ -{-2(1^ -3 a -\- 20. 

3.  3ar^4-ar^H-3a;  +  5  and  3a^ +  7 a^ -x  +  W. 

4.  2a^-{-2a'b-ab^  +  6b^  and  2a^ -Qa^b  +  T ab^-6b^  ^ 

5.  2a^-10^  +  24anda^  +  a;2_3^_^9^ 

6.  a*  +  ot^  + 1  and  a*  —  a^-i- 1. 

7.  a^4-a-2anda*  +  a3-2a-4. 

8.  a^  +  ab'--2b^2iiida'-\-a^b-2ab^-4:b\ 

9.  a«-6a*  +  lla='-6anda«-9a*  +  26a2-24. 

10.  a^-2a-3,a^  +  a'-4:a-4:,a^-7a-6. 

11.  a3-2a2-5a  +  6,  a8-13a  +  12,  2a8-lla2  +  18a-9. 

IV.    CUBE  ROOTS  OF  POLYNOMFALS  AND 
ARITHMETICAL  NUJVIBERS 

1.  By  considering  that  the  cube  root  of  a^  4-  3  a^b  +  3  a6^  +  b^ 
is  a  +  b,  and  employing  a  method  similar  to  the  one  used  for 
square  roots,  it  can  be  shown  that, 

TJie  first  term  a  of  the  root  is  the  cube  root  of  the  first  term  a\ 
TJie  second  term  of  the  root  can  be  obtained  by  dividing  3  a*6 
by  3  a\  the  so-called  trial  divisor. 


510  ADVANCED   ALGEBRA 

The  complete  divisor  is  obtained  by  dividing  the  last  three 
terms  by  b,  i.e.  the  complete  divisor  is  3  a^  +  3  a6  +  &l 
The  work  may  be  arranged  as  follows : 

a' 


Trial  divisor  =  3  a^ 

Complete  divisor  =  3  a^ -\- 3  ab -\- b^ 


3a26-f3a62-f-53 
3  a^ft  +  3ab'  -f  b^ 


Ex.  1.     Extract  the  cube  root  of  8  a;^  -  27  2/^  -  36  a^2/  +  54  xy^. 

Arranging,  8 x^ -  36 x'^y  +  54 xy"^ - 27 y^  |  2a;-3y 

8x3 


3a2                    =12x2 

Sab  +  b^            =          -18x2/  +  9?/2 

-36x2y  +  54xy2_27y8 

3a24-3a6  +  62  =  12x2-18xy  +  92/2 

-36x2?/  +  54x?/2_27y8 

EXERCISE 

Extract  the  cube  root  of  the  following  expressions : 

1.  ar^  +  6aj2  +  12a;  +  8. 

2.  S  a' -  S4:  a'b-h  294:  ab^- 3^3  b\ 

3.  a«  -  6  a^6  +  12  ttV- 8  a^^^ 

5.  216  a«  -  756  a^6  +  882 a^b^  -  343  b\ 

6.  l+3«  +  6a2  +  7a3  +  6a*  +  3a^  +  a«. 

7.  8aj«-36a^  +  66ic*-63a.-3  4-33a^-9a;  +  l. 

8.  30x^-12x-{-S-25a^-{-30x'-12a^  +  Safi, 
9  S  ar'^^  +  36  x-Py''+^  +  54  o^V^^  +  27  /«+». 

10.    8a^-60a^  +  114a^  +  55a«-171a2-135a-27. 


APPENDIX  511 

2.   The  cube  root  of  an  arithmetical  number  is  found  in  a 
similar  manner. 

Ex.  1.     Extract  the  cube  root  of  6644.672. 

Commencing  at  the  decimal,  divide  the  number  into  groups  of  digits, 
each  group  to  contain  3  digits. 

6'644'.672  |  18.8 
1 


8a2=300 

6644 

3  aft  =  240 

6^=   64 
604 

4832 

8.180-^  =  97200 

812672 

3.180.8=  4320 

82=       64 

101584 

812672 

EXERCISE 
Extract  the  cube  roots  of  the  following  numbers: 

1.  12167.  5.   830584.  9.   .059319. 

2.  9261.  6.   531441.  10.   704969. 

3.  39304.  7.   32.768.  11.   43243551. 

4.  175.616.  8.    .274625.  12.   237176659. 

Find  to  two  decimal  places  the  cube  roots  of  the  following 
numbers : 

13.   2.  14.   5.  15.   8.  16.   10. 


612  ADVANCED  ALGEBRA 

V.     INDETERMINATE   EQUATIONS  OF   THE  FIRST 
DEGREE 

1.  It  was  shown  in  §  189  that  a  linear  equation  involving 
two  unknown  quantities,  such  as  'dx-\-2y  =  lj  has  an  infinite 
number  of  solutions.  Similarly  any  system  of  equations  is 
indeterminate,  if  the  number  of  unknown  quantities  is  greater 
than  the  number  of  equations.  By  introducing  the  condition 
that  the  roots  shall  be  positive  integers,  the  number  of  solu- 
tions can  frequently  be  limited. 

2.  If  the  equation  ax  +  by=c  (1) 

is  satisfied  by  the  values  x  =  ri,  and  ^=7*2,  then  it  is  also  sat- 
isfied by  the  values : 

x  =  ri-\-inbt  ( x  =  i\  —  mb, 

or      J 
y  =  ?  2  —  ma^  [  y  =  rg  -f  ma. 

For,  substituting  in  (1),  we  obtain 

a(ri  ±  mh)  +  6  (rg  T  ma)  =  c. 

I.e,  ari-{-br2==c. 

3.  If  one  set  of  integers  is  known  that  satisfies  an  indeter- 
minate equation  involving  two  unknown  quantities,  all  solu- 
tions can  be  found  by  the  preceding  paragraph. 

Thus,  the  equation  Sx  +  2y  =  M  is  satisfied  by  the  values  x  =  10, 
y  =  2.     Hence  it  is  satisfied  by  any  values  of  the  form : 

X  =  10  -  2  w, 

y=   2  +  Sm. 

E.g.     If  7)1  =  1,  01  =  8,  and  y  =  5. 

If  w  =  2,  x  =  6,  and  y  =  S,  etc. 

Ex.  1.    Solve  in  positive  integers : 

7x-\-16y  =  209.  •      (1) 


APPENDIX  613 

Dividing  both  members  by  the  smaller  coefficient,  i.e.  7, 

a;  +  2y  +  ^  =  29  +  ^.  (2j 

Since  x  and  y,  and  hence  x-\-2y,  are  to  be  integers,  the  fractional 
parts  of  the  two  members  must  either  be  equal,  or  differ  by  an  integral 
number. 

Assume  the  first  of  these  cases,  if  it  produces  an  integral  y ; 

?i!=|,    i.e.y  =  8. 

Substituting  in  (2) ,       x-\-6  =  29,    i.  e.x  =  23. 

Hence  if  m  denotes  any  integer,  the  general  solutions  are  (§  3)  ; 

x  =  23-16»»,  .  (3) 

y=   3+   7  m.  (4) 

Restricting  the  answers  to  positive  values, 
we  have  from  (3),  w^l) 

from  (4),  m^O. 

Therefore  m=   0,  or    1. 

Hence  .  aj  =  23,  or    7, 

and  y  =   3,  or  10. 

Ex.  2.    Solve  in  positive  integers: 

43x4-20  2/ =  798.  (1) 

(2) 


Dividing  by  20, 

2x-\-y  +  ^  =  S9  + 

18 
20 

Hence 

Bx 
20 

:1^,     i.e.x 
20 

=  6 

Substituting  in  (2), 

12  +  2/ 

=  39,  i.e.  y 

=  2 

Hence  the  general  solutions  are  (§  3)  : 

x  = 

:     6  +  20W, 

y  = 

:  27  -  43  m. 

To  obtain  positive  values  for  x  and  y,  in  must  be  zero. 
Hence  x  =  6  and  y  =  21  are  the  only  solutions. 
2l 


514  ADVANCED  ALGEBRA 

Ex.  3.    Solve  in  positive  integers  : 

23  CC  + 37  2/ =  3000.  (1) 

Dividing  by  23,  ^  +  ^ +  ^  =  ^^^"^H*  ^^^ 

Equating  the  fractional  parts  of  the  two  members  produces  a  fractional 

y,  hence  let 

— ^  = (-  w,  where  n  is  an  integer. 

23       23  ^ 

Or  Uy  =  10  +  23  n. 

This  is  another  indeterminate  equation,  but  a  simpler  one  than  (1). 
In  complex  cases,  this  equation  may  be  treated  by  the  regular  method, 
while  in  simpler  ones  we  find  by  trial  a  value  of  n  which  produces  an  inte- 
gral y.    Evidently,  if  w  =  2,  10  +  23  w  is  divisible  by  14.     Or,  if 

w  =  2, 

2/ =  4. 

Equation  (2)  may  be  written, 

x  +  y  +  7i  =  130. 

Substituting,  a;  +  4  +  2  =  130,  i.  e.  x  =  124. 

Hence  the  general  solutions  are  : 

X  =  124  -  37  m,  (3) 

y=     4  +  23  m.  (4) 

To  make  x positive,  w^3;  to  make  y  positive,  m^O. 
Hence  we  have  the  following  possible  values  : 

w  =     0,     1,     2,    3. 

x  =  124,  87,  50,  13. 

y=     4,  27,  50,  73. 

Ex.  4.    Solve  in  positive  integers : 

6x-17y  =  27, 

Dividing  by  6,  a;-3y  +  |  =  4  +  ^. 

6  o 


APPENDIX  516 

Hence  y  =  3, 

and  aj  — 9  =  4,  i.e.  a;  =  13. 

Hence  the  general  solutions  are  : 

a;  =  13-(-17)m, 
or  a;  =  13  +  17  wi, 

and  y  =  3  +  6  m. 

By  assigning  to  m  any  positive  integral  value,  we  obtain  an  unlimited 

number  of  solutions :  ^      ,      «      o 

m=   Oj     1,     2,     3,  ..., 

a;  =  13,  30,  47,  64,  ..., 

2/=  3,    9,  15,-  21,  .... 

EXERCISE 
Solve  in  positive  integers : 

1.  3a;  +  72/  =  26.  10.  2 a:  + 13 2/ =  37. 

2.  2a;  +  92/  =  27.  11.  7 a;  +  48 y  =  1000. 

3.  7a;  +  162/  =  62.  12.  3a;  +  82/=100. 

4.  7a;4-ll2/  =  151.  13.  18  a;  +  7  y  =  600. 

5.  38a;  +  72/  =  216.  14.  5a;  +  32/  =  70. 

6.  9a;  +  292/=179.  15.  6 a;  + 17 y  =  500. 

7.  47  a;  + 11 2/ =  1117.  16.  3  a;  +  H  2/ =  100. 

8.  4a;  +  72/  =  99.  17.  5a;  +  72/-f 4  =  56. 

9.  16a;  +  72/  =  601.  18.  46  a;  +  41 2/ =  2188. 

Solve  in  least  possible  positive  integers : 

19.  lla;-22/  =  7.  23.   3a;-82^  =  10. 

20.  13  a; -4  2/ =  19.  24.    5  a; -7  2/ =  3. 

21.  17a;-52/  =  24.  25.   7a;-182/  =  20. 

22.  12  a;  — 5  2/  =  1.  26.   14  a?  —  45  2/ =  11. 


516  ADVANCED  ALGEBRA 

Solve  in  positive  integers : 

27.     i  28. 


4a;-52/-62  =  -66.  [^.x-Sy  ■\-2z^l4.. 

29..  Divide  142  into  two  parts  such  that  one  part  is  a  mul- 
tiple of  9,  the  other  a  multiple  of  14. 

30.  Divide  1591  into  two  parts  such  that  one  part  is  a  mul- 
tiple of  23,  the  other  of  34. 

31.  Find  two  fractions  whose  denominators  are  respectively 
7  and  3,  and  whose  sum  equals  |-f-. 

32.  In  what  manner  can  $  15  be  paid  in  five-dollar  bills  and 
two-dollar  bills  ? 

33.  A  farmer  sold  a  number  of  horses  and  cows  for  $447, 
receiving  $112  for  each  horse  and  $37  for  each  cow.  How 
many  did  he  sell  of  each  ? 

34.  A  grocer  bought  a  number  of  pounds  of  tea  and  coifee 
for  $5.10,  paying  for  tea  40^  per  pound,  and  for  coffee  19^  per 
pound.     How  many  pounds  did  he  buy  of  each  ? 

35.  A  farmer  sold  a  number  of  cows,  sheep,  and  pigs  for 
$  140,  receiving  $  31  for  each  cow,  $  11  for  each  sheep,  and  $  9 
for  each  pig.  How  many  did  he  buy  of  each,  if  the  total 
number  of  animals  was  10  ? 


yi.    YARIATION" 

1.  Direct  variation.  When  the  ratio  of  two  variables  is  con- 
stant, each  variable  is  said  to  vary  directly  as  the  other.     Thus, 

X  varies  directly  as  y  (or  briefly,  x  varies  as  y)^  if  -  =  m,  a 
constant.  ^ 

E.g.  The  weight  of  a  quantity  of  water  varies  as  its  volume.  The 
distance  traversed  by  a  man  walking  at  a  uniform  rate  varies  as  the  time 
during  which  he  walks,  etc. 


APPENDIX  517 

2.  The  sign  of  variation  is  oc.     It  is  read  "varies  as." 
Thus,  if  -  =  3,  then  x  cc  y. 

y 

3.  li  X  oc/,  then  x  =  my,  where  m  is  a  constant.     For  by 
definition,  -  =  a  constant.     Let  this  constant  be  m,  then  x  =  my, 

y 

Note.     In  the  present  chapter  the  letters  m,  n  denote  constants.     If  x 

varies  as  ?/,  then  -  is  always  taken  equal  to  m. 

y  , 

Ex.  1.    li  xccy,  and  a;  =  3,  when  2/ =  4,  find  the  value  of  x 

when  2/ =  10. 


Since 
len 

y 

X  =  S,  and  y  =  4,  then  w  =  f . 

Hence 

x  =  iy. 

When  y  = 

10, 

ic  =  f  .  10  33  7^. 

4.  Inverse  variation.  One  quantity  is  said  to  vary  inversely 
as  another  if  it  varies  as  the  reciprocal  of  the  other.  Thus,  x 
varies  inversely  sls  y,  ii  xcc  -• 

5.  li  X  varies  inversely  as/,  then  xy  is  a  constant. 

Let  x  =  m  '  -,  where  7n  is  a  constant. 
y  .  .  . 

Hence  we  have  xy  —  m. 

6.  Joint  variation.  One  number  is  said  to  vary  jointly  as  a 
number  of  others  if  it  varies  as  their  product. 

Thus,  X  varies  jointly  as  y  and  ziixc/:  yz,  or  if  x  =  jnyz. 

Ex.  2.  The  volume  v  of  a  circular  cone  varies  jointly  as  its 
height  h,  and  the  square  of  the  radius  r  of  its  base.  When 
r  =  10,  and  h  =  12,  then  v  =  1256.     Find  v,\ih^  5  and  r^  5. 

Let  X  be  the  required  volume. 

Since  v  =  mhr'^,  (1) 

we  have  1256  =  m  •  12  •  100,  (2) 

and  X  =  ?w  •  5  •  26.  (3) 


518  ADVANCED  ALGEBRA 

We  may  find  the  value  of  m  from  (2),  and  substitute  it  in  (3).     It  is, 
however,  simpler  to  divide  the  members  of  (3)  by  those  of  (2). 

X         5-25 


1256     12 .  100 
Therefore  x  =  130f . 

EXERCISE 

1.  If  a;  oc  y,  and  a;  =  6,  when  y  =  2j  find  x  when  ?/  =  7. 

2.  If  a;  QC  y,  and  x  =  l,  when  y  =  S,  find  x  when  y  =  2. 

3.  If  X  oc  -,  and  a;  =  2,  when  y  =  6,  find  x  when  y  =  S. 

4.  If  a;  oc  -,  and  a;  =  7,  when  2/  =  4,  find  x  when  y  =  10. 

y 

5.  If  aj  cc  -,  and  x  =  9,  when  y  =  2.  find  a;,  if  a;  =  v. 

6.  If  x  varies  jointly  as  ?/  and  z ;  and  a;  =  120,  when  2/  =  2, 
z  =  15 ;  find  x  when  y  =  4,  z  =  17. 

7.  If  a;  0C-,  and  a;  =  16,  when  2/ =  3,  find  a;  when  i/  =  6. 

y 

8.  If  a;  oc  - ,  and  a;  =  4  a^,  when  2/  =  6^,  find  x  when  2/  =  4  aft. 

9.  If  a;  oc  2/^,  and  a;  =  32,  when  ?/  =  8,  find  x  when  2/  =  5. 

10.  If  XOC-,  and  a;  =  2,  when  2/ =  12,  z  =  2',  find  a;  when 
2/  =  7,  z  =  S.    ^ 

11.  If  a;  oc  y,  and  2/  »^  2;,  prove  that  xccz. 

12.  If  a;  oc  -,  and  y  (f:-,  prove  that  a;  oc  2. 

2/  z 

13.  If  a;  oc  ?/,  prove  that  x"^  oc  2/^. 

14.  If  2  a  +  3  6  oc  3  a  -f  6,  and  a  =  2,  if  6  =  1,  find  a  in  terms 
of  b. 


APPENDIX  519 

15.  If  X  varies  directly  as  y^  and  inversely  as  2?,  and  05  =  2, 
when  y  =  ly  z  =  3;  find  the  value  of  x,  when  2/  =  2,  2;  =  4. 

16.  It  xcc^,  and  a;  =  2,  when  2/  =  12, 2  =  2 ;  find  a;  when  y  =  7, 
z  =  Z.  "■ 

17.  The  area  of  a  circle  varies  as  the  square  of  its  radius. 
If  the  area  of  a  circle  is  100  square  yards,  when  the  radius  is  a 
feet,  find  the  area  of  a  circle  whose  radius  is  3  a  feet. 

18.  The  volume  of  a  sphere  varies  as  the  cube  of  its  diam- 
eter. If  the  volume  of  a  sphere  is  10  cubic  inches,  when  its 
diameter  is  a  inches,  find  the  volume  of  a  sphere  whose  diameter 

is  ^  inches. 

19.  The  volume  of  a  gas  varies  as  the  absolute  temperature, 
and  inversely  as  the  pressure.  The  volume  of  a  body  of  gas  is 
100  cubic  inches  when  the  pressure  is  20  and  the  absolute  tem- 
perature 250.  What  will  be  the  volume  when  the  pressure  is 
30  and  the  absolute  temperature  360  ? 

20.  The  illumination  of  an  object  varies  inversely  as  the 
square  of  the  distance  from  the  source  of  light.  If  the  illu- 
mination of  an  object  at  a  distance  of  5  feet  from  a  source  of 
light  is  10,  what  is  the  illumination  at  a  distance  of  15  feet  ? 

VII.    LOGARITHMS 

1.  If  &"  =  n,  (1) 
then  X  is  the  logarithm  of  n  to  the  base  6,  or  in  symbols 

X  =  logb  n.  (2) 

Thus,  since  28  =  8,     3  =  logs  8 ; 

since  34  =  81,  4  =  logs  81. 

2.  The  two  equations  (1)  and  (2)  express  the  same  relation, 
and  whenever  we  are  unable  to  solve  a  problem  stated  in  the  form 


520  ADVANCED  ALGEBRA 

(2),  we  transform  it  to  the  form  (1),  i.e.  we  state  it  as  an  exponen- 
tial equation. 

Ex.  1.    Determine  a;  if  a;  =  logs  125. 

Writing  this  equation  in  the  exponential  form,  we  have 

5*  =  125. 
Hence  x  =  S. 

Ex.  2. .  Eind  log«  1. 

Let  X  =  loga  1,  then  a''  =  1. 

Hence  « =  0,       or  loga  1  =  0. 

3.    Logarithms  to  the  base  10  are  known  as  common  loga- 
rithms.    If  no  base  is  written,  10  is  understood  as  the  base. 

Thus,  log  20  =  logio  20,  loga  =  logioa. 

Ex.  3.   Eind  log  100. 

Let  X  =  log  100  ;  then  10*  =  100. 

,  Hence  x  =  2,  or  log  100  =  2. 

Note.    The  definition  of  logarithm  can  be  expressed  by  the  equation 

EXERCISE 

Express  the  following  relations  as  logarithmic  equations : 

1.  102  =  100.  4.    (V2)2  =  2.  7.   10-2  =  ^^. 

2.  103  =  1000.  5.   72  =  49.  8.    (V3/  =  9. 

3.  10«  =  1.  6.   2^  =  32. 

Express  the  following  relations  as  exponential  equations : 
9.    3  =  log4  64.  11.    -3  =  log.001.    13.    -3  =  log2|c 

10.    6 =logio  1000000.     12.    a;  =  log  a.  14.    log  1  =  0. 


APPENDIX 


521 


Determine  the  following  logarithms: 


15.  log  1000. 

16.  log  10000. 

17.  loga  a. 

18.  log2  64. 

19.  log.l. 

Simplify : 

29.    2  log2 10  +  logs  1. 


20.  log^f 

21.  log  100000. 

22.  log3  243. 

23.  log^^^. 

24.  log -Wo. 


25.  logi6  4. 

26.  logar  3. 

27.  log  .00001. 

28.  log  10000000. 


Solve  the  following  equations 

31.  log3Cc  =  5. 

32.  log8ic  =  2. 


30.    log5  52  4.1ognll. 


33.    lo,sr,16  =  2. 


34.   Iog^l0000  =  4. 


4.    The  logarithm  of  1  to  any  base  is  0. 

Let  X  =  logj  1,  then  &*  =  1,  hence  a;  =  0. 

6.    The  logarithm  of  the  base  is  1. 

Let  X  =  logj  b,  then  6*  =  b,  hence  x  =  l. 

6.    The  logarithm  of  a  product  is  equal  to  the  sum  of  the  loga- 
rithms of  its  factors. 

I.e.    if  log,,  m  —  x,  and  log,,  n  =  yf  (1) 

then  \ogi,mn  =  x  +  y.  (2) 

Writing  these  equations  in  the  exponential  form  (§  2),  we 
have : 

If  m  =  6*,  and  w  =  fe*,  then  mn  =  6*+*. 

But  in  this  form  the  conclusion  is  obvious.     Substituting  the 
values  of  x  and  y  in  equation  (2),  and  omitting  the  bases,  we 


have 


log  (mn)  =  log  m  +  log  n. 


th£n  \og^-  =  x-y,  (2) 


522  ADVANCED  ALGEBRA 

E.g.  log  16  =  log  (3x5)=  log  3  +  log  5. 

log  (abc)  =  log  a  4-  log  {be)  =  log  a  +  log  b  +  log  c. 

7.  27ie  logarithm  of  a  quotient  is  equal  to  the  logarithm  of 
the  dividend  diminished  by  the  logarithm  of  the  divisor. 

I.e.  if  logfc  m  =  x,  and  logj  n  =  y,  (1) 

n 

This  really  means  (§  2)  : 
If  m  =  6"=  and  n  =  6^,  then  —  =  b*-».  (3) 

But  in  this  form  the  conclusion  is  obvious.     Substituting  the 
values  of  x  and  y  in  (2),  and  omitting  the  bases,  we  obtain 

log  —  =  log  m  —  log  n, 
n 

E.g.  log  f  =  log  5  -  log  2. 

logl^  =  log^  =  log3  +  log5-log2. 
log  5  =  log  ^^  =  log  10  -  log2  =  1  -  log  2. 

8.  The  logarithm  of  a  power  of  a  number   is  equal   to  the 
logarithm,  of  that  number  multiplied  by  the  exponent  of  the  power. 

I.e.  if  logj  n  =  Xy  then  logj  n^  —px.  (1) 

This  means  (§  2)  : 

If  n  =  ¥,  then  n^  =  ¥'.  (2) 

Again  the  conclusion  is  obvious. 

Substituting  the  value  of  x  in  equation  (1),  and  omitting  the 
bases,  we  have  \o^„P=p\o^n. 

E.g.  log 2^  =  10 log 2  ;  log i  =  log 2-3  =  -3 log 2  ; 

log^^  =  log2  +  31og7-51og2; 

log  2i2i32  =  3  log  2  +  2  log  3  -  5  log  5  -  2  log 6. 
6^  X  6^ 


APPENDIX  523 

9.    The  logarithm  of  a  root  of  a  number  is  equal  to  the  loga- 
rithm of  the  number  divided  by  the  index. 

log  ^n  =  log  /i"  =  ?i  log  n.     (§  8) 

P 

E.g.  log  ^7  =  I  log  7. 

Iog2ili|  =  31og2  +  ilog3-21og5-|log2. 

52.^2 

Ex.  1.   Given  log  2  =  .30103,  and  log  3  =  .47712,  find  log  288. 
log  288  =  log  32  .  25  =  2  log  3  +  5  log  2. 

2  log  3=    .95424 

5  log  2  =  1.50515 

Therefore  log  288  =  2. 45939 

Ex.  2.    Given  log  2  =  .30103,  find  log  25. 

log  25  =  2  log   5  =  2  log  (Y)  =  2  (log  10  -  log  2). 

But  log  10  =  1.00000         (§  5) 

lose   2=    .30103 


log  10  -  log   2  =    .69897 
Therefore  log  25  =  1.39794. 


EXERCISE 
Given 


log  2  =  .30103,  log  3  =  .47712,  log  5  =  .69897,  log  7  =  .84510, 
find  the  values  of  the  following  logarithms : 


1. 

log  6. 

8. 

log  2000. 

15. 

log  4900. 

2. 

log  14. 

9. 

log  729. 

16. 

log  225. 

3. 

log  21. 

10. 

log-V- 

17. 

lQg\V. 

4. 
5. 

log  12. 
logf. 

11. 
12. 

log  ^2. 
log  36. 

18. 

log^"?/^^. 

6. 

log  8. 

13. 

log  720. 

19. 

log2i. 

7. 

log  30. 

14. 

log  343. 

20. 

log  ^16. 

•^24  ADVANCED  ALGEBRA 

21.    W^.  24.    log 2'^. 

22.  log^iii^iiJ^.        ''•  ^'°'^^^- 

— ^^  26     Wl552 

23.    logA/4^.  •       ^.003^' 

Express  in  terms  of  log  a,  log  b,  log  c,  and  log  c? : 
27.    log(aW).  ^^^    logV«liI\ 


aWbVd 


VcH 


29.  log^   ^^  ^^\  ^       10r/'»/>V 

^  c*  33.    logii^^^^. 

-^(^ 

30.  log:^.  3,.  logl50^^ 

Simplify : 

35.  log^+log^+log^-2log5c. 

36.  log  (x  +  2/)  4-  log  (x-y)-  log  (a^^  -  /). 

37.  logaH-log6  +  log-  +  log^-loga6. 

c  a 

38.    logi-%a  +  log|f  39.    logjgj's-^ 

Solve  the  following  equations  : 
40.'   3  log  a;  =  2  log  8. 

41.  log  ar^  —  log  a;  =  log  — 

X 

42.  log  35  —  log  a;  =  log  7. 

43.  logl6a;-log8a^  =  log8a;2-21og4a;. 


APPENDIX  525 

10.  Properties  of  common  logarithms.  Since  only  a  few 
numbers  are  perfect  powers  of  10,  it  is  evident  that  common 
logarithms  in  general  are  not  integral  numbers. 

The  characteristic  is  the  integral  part  of  a  logarithm.  The 
mantissa  is  the  fractional  part  of  a  logarithm. 

E.g.  in  log 200  =  2.30103,  the  characteristic  is  2,  and  the  mantissa  is 
.30103. 

11.  The  values  of  logarithms  are  usually  so  written  that  the 
mantissa  is  positive. 

E.g.  the  value  of  log  .03  =  -  1.52288.  But  this  equals 
-2 +  .47712. 

This  is  usually  written : 

log  .03  =  .47712  -  2 ;  or,  log  .03  =  8.47712  - 10. 

Note.  Some  writers  place  the  negative  sign  over  the  characteristic,  to 
indicate  that  the  characteristic  alone  is  to  be  taken  negative,  thus  log  .03 
may  be  written  2.47712. 

12.  The  characteristic  of  the  logarithm  of  any  number  can  be 
written  by  inspection. 

(a)  W  =  l,        hence  logl       =0. 

10^  =  10,      hence  log  10     =1. 

10^  =  100,     hence  log  100    =2. 

10^  =  1000,  hence  log  1000  =  3,  etc. 

Therefore  the  logarithm  of  any  number 

between  1  and  10  =  0  +  a  fraction ; 
between  10  and  100  =  1  +  a  fraction ; 
betwe^  100  and  1000  =  2  -f  a  fraction ;  etc. 


626  ADVANCED  ALGEBRA 

Hence  it  follows  that : 

The  characteristic  of  the  logarithm  of  any  number  greater  than 
unity  is  positive,  and  is  less  by  one  than  the  number  of  digits  in 
the  integral  part. 

Thus,  the  characteristic  of  log  47. 127  is  1,  of  log  47216.2  is  4,  etc. 
(f>)  Since    10«   =  1,         we  have  log  1=0. 

Since    10~^  =  yijy,       we  have  log  .1        =  —  1. 

Since     10"^  =  yi^,     we  have  log  .01      =  —  2. 

Since    10~^  =  yoVu'  ^®  ^^y^  log  .0001  =  —  3,  etc. 


Hence  the  characteristic  of  the  logarithm  of  a  number  less 
than  unity  is  negative.  If  no  cipher  immediately  follows  the 
decimal  point,  the  number  lies  between  1  and  .1,  hence  its 
logarithm  =  —  1  +  a  fraction.  If  one  cipher  immediately  fol- 
lows the  decimal  point,  the  number  lies  between  ,1  and  .01, 
hence  its  logarithm  =  —  2  +  a  fraction,  etc. 

Hence  we  have  in  general : 

If  a  number  less  than  unity  be  expressed  as  a  decimal,  the 
characteristic  of  its  logarithm  is  negative  and  numerically  greater 
by  one  than  the  nu7tiber  of  ciphers  immediately  following  the 
decimal  point. 

E.g.  the  characteristic  of  log  .476  is  -1,  of  log  .0123  is  -2,  of  log  .000047 
is  —  5. 

13.  If  two  numbers  differ  only  in  the  position  of  the  decimal 
point,  their  logarithms  have  equal  mantissas,  but  different  charac- 
teristics. 

Consider  the  numbers  4.1456  and  .0041456. 

4.1456  =  1000  X  .0041456. 

Hence  log  4.1456  =  log  1000  +  log  .0041456 

=  3 +  log  .0041456.  • 


APPENDIX  527 

Hence  the  two  logarithms  differ  by  an  integral  number,  i.e. 
their  mantissas  are  equal. 

14.  The  table  of  logarithms  on  pages  528  and  529  is  only  a 
table  of  mantissas,  since  the  characteristics  may  be  found  by 
inspection. 

15.  To  find  the  logarithm  of  a  number.  First  write  the  charac- 
teristic by  inspection.  In  determining  the  mantissa  do  not 
consider  the  decimal  point.  The  mantissa  is  found  from  the 
table  as  follows  : 

(a)  Let  the  given  number  consist  of  three  figures,  as  72.3. 
In  the  column  headed  N  look  for  the  first  two  significant  figures 
{i.e.  72).  The  required  mantissa  is  then  found  on  the  same  hor- 
izontal line,  in  the  column  headed  by  the  third  figure  {i.e.  3). 

Thus,  on  a  line  with  72,  in  the  column  headed  3,  we  find  8591. 

As  the  characteristic  is  1,  we  have 

log  72.3  =  1.8591. 

(6)  If  the  number  consists  of  less  than  three  figures,  add 
ciphers ;  thus,  to  find  log  6,  find  log  6.00,  log  .57  =  log  .570,  etc. 

(c)  Let  the  given  number  consist  of  more  than  three  figures, 
as  .064345. 

The  mantissa  of  log  64300  =  .8082. 

The  mantissa  of  log  64400  =  .8089. 

That  is,  if  the  number  increases  100,  its  logarithm  increases 
.0007. 

Hence,  if  the  number  64300  increases  45,  its  logarithm  will 
increase  y^^  of  .0007,  or  .0003. 

Hence  the  mantissa  =  .8082  -f  .0003  =  .8085. 

Whence,   log  .064345  =  .8085  -  2  or  8.8085  - 10. 

16.  The  difference  between  two  consecutive  mantissas  in 
the  table  is  called  the  tabular  difference. 


528 


ADVANCED  ALGEBRA 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

lO 

0000 

0043 

0086 

0128 

0170 

0212 

0253 

0294 

0334 

0374 

II 

0414 

0453 

0492 

0531 

0569 

0607 

0645 

0682 

0719 

0755 

12 

0792 

0828 

0864 

0899 

0934 

0969 

1004 

1038 

1072 

1 106 

13 

1139 

^^73 

1206 

1239 

1271 

1303 

1335 

1367 

1399 

1430 

14 

1461 

1492 

1523 

1553 

1584 

1614 

1644 

1673 

1703 

1732 

15 

1761 

1790 

1818 

1847 

1875 

1903 

1931 

1959 

1987 

2014 

16 

.2041 

2068 

2095 

2122 

2148 

2175 

2201 

2227 

2253 

2279 

17 

2304 

2330 

2355 

2380 

2405 

2430 

2455 

2480 

2504 

2529 

18 

2553 

2577 

2601 

2625 

2648 

2672 

2695 

2718 

2742 

2765 

19 

2788 

2810 

2833 

2856 

2878 

2900 

2923 

2945 

2967 

2989 

20 

3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

21 

3222 

3243 

3263 

3284 

3304 

3324 

3345 

3365 

3385 

3404 

22 

3424 

3444 

3464 

3483 

3502 

3522 

3541 

3560 

3579 

3598 

23 

3617 

3636 

3655 

3674 

3692 

3711 

3729 

3747 

3766 

3784 

24 

3802 

3820 

3838 

3856 

3874 

3892 

3909 

3927 

3945 

3962 

25 

3979 

3997 

4014 

4031 

4048 

4065 

4082 

4099 

4116 

4133 

26 

4150 

4166 

4183 

4200 

4216 

4232 

4249 

4265 

4281 

4298 

27 

4314 

4330 

4346 

4362 

4378 

4393 

4409 

4425 

4440 

4456 

28 

4472  j  4487 

4502 

4518 

4533 

4548 

4564 

4579 

4594 

4609 

29 

4624 

4639 

4654 

4669 

4683 

4698 

4713 

4728 

4742 

4757 

30 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

31 

4914 

4928 

4942 

4955 

4969 

4983 

4997 

501 1 

5024 

5038 

32 

5051 

5065 

5079 

5092 

5105 

5119 

5132 

5145 

5159 

5172 

33 

5^^5 

5198 

5211 

5224 

5237 

5250 

5263 

5276 

5289 

5302 

34 

5315 

5328 

5340 

5353 

5366 

5378 

5391 

5403 

5416 

5428 

35 

5441 

5453 

5465 

5478 

5490 

5502 

55H 

5527 

5539 

5551 

36 

^IP 

5575 

5587 

5599 

5611 

5-623 

5635 

5647 

5658 

5899 

37 

5682 

5694 

5705 

5717 

5729 

5740 

5752 

5763 
5877 

5775 

38 

5798 

5809 

5821 

5832 

5843 

5855 

5866 

5888 

39 

5911 

5922 

5933 

5944 

5955 

5966 

5977 

5988 

5999 

6010 

40 

6021 

6031 

6042 

6053 

6064 

6075 

6085 

6096 

6107 

6117 

41 

6128 

6138 

6149 

6160 

6170 

6180 

6191 

6201 

6212 

6222 

42 

6232 

6243 

6253 

6263 

6274 

6284 

6294 

6304 

6314 

6325 

43 

6335 

6345 

6355 

6365 

6375 

6385 

6395 

6405 

6415 

6425 

44 

6435 

6444 

6454 

6464 

6474 

6484 

6493 

6503 

6513 

6522 

45 

6532 

6542 

6551 

6561 

6571 

6580 

6590 

6599 

6609 

6618 

46 

6628 

6637 

6646 

6656 

666s 

6675 

6684 

6702 

6712 

47 

6721 

6730 

6739 

6749 

6758 

6767 

6776 

6785 

6794 

6803 

48 

6812 

6821 

6830 

6839 

6848 

6857 

6866 

6875 

6884 

6893 

49 

6902 

6911 

6920 

6928 

6937 

6946 

6955 

6964 

6972 

6981 

50 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 

51 

7076 

7084 

7093 

7101 

7110 

7118 

7126 

7135 
7218 

7143 

7152 

52 

7160 

7168 

7177 

7185 

7193 

7202 

7210 

7226 

7235 

53 

7243 

7251 

7259 

7267 

7275 

7284 

7292 

7300 

7308 

7316 

54 

7324 

7332 

7340 

7348 

7356 

7364 

7372 

7380 

7388 

7396 

N 

0 

1 

~2~ 

3 

4 

5 

6^ 

7 

8 

9 

APPENDIX 


529 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

55 

7404 

7412 

7419 

7427 

7435 

7443 

7451 

7459 

7466 

7474 

56 

7482 

7490 

7497 

7505 

7513 

7520 

7528 

7536 

7543 

7551 

57 

7559 

7566 

7574 

7582 

7589 

7597 

7604 

7612 

7619 

7627 

58 

7634 

7642 

7649 

7657 

7664 

7672 

7679 

7686 

7694 

7701 

59 

7709 

7716 

7723 

7731 

7738 

7745 

7752 

7760 

7767 

7774 

6o 

7782 

7789 

7796 

7803 

7810 

7818 

7825 

7832 

7839 

7846 

6i 

7^53 

7860 

7868 

7875 

7882 

7889 

7896 

7903 

7910 

7917 

62 

7924 

7931 

7938 

7945 

7952 

7959 

7966 

7973 

7980 

7987 

63 

7993 

8cxx> 

8007 

8014 

8021 

8028 

8035 

8041 

8048 

8055 

64 

8062 

8069 

8075 

8082 

80S9 

8096 

8102 

8109 

8116 

8122 

65 

8129 

8136 

8142 

8149 

8156 

8162 

8169 

8176 

8182 

8189 

66 

8195 

8202 

8209 

8215 

8222 

8228 

8235 

8241 

8248 

8254 

67 

8261 

8267 

8274 

8280 

8287 

8293 

8299 

8306 

8312 

8319 

68 

8325 

8331 

^32>^ 

8344 

8351 

8357 

8363 

8370 

8376 

8382 

69 

8388 

8395 

8401 

8407 

8414 

8420 

8426 

8432 

8439 

8445 

70 

8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500 

8506 

71 

8513 

8519 

8525 

8531 

8537 

8543 

8549 

8555 

8561 

8567 

72 

8573 

8579 

8585 

8591 

8597 

8603 

86c9 

8615 

8621 

8627 

73 

8633 

8639 

8645 

8651 

8657 

8663 

8669 

8675 

8681 

8686 

74 

8692 

8698 

8704 

8710 

8716 

8722 

8727 

8733 

8739 

8745 

75 

8751 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

76 

8808 

8814 

8820 

8825 

8831 

8837 

8842 

8848 

8854 

8859 

77 

8865 

8871 

8876 

8882 

8887 

8893 

8899 

8904 

8910 

8915 

78 

8921 

8927 

8932 

8938 

8943 

8949 

8954 

8960 

8965 

8971 

79 

8976 

8982 

8987 

8993 

8998 

9004 

9009 

9315 

9020 

9025 

80 

9031 

9036 

9042 

9047 

9053 

9058 

9063 

9369 

9074 

9079 

81 

9085 

9090 

9096 

9IOI 

9106 

9112 

9117 

9122 

9128 

^^il 

82 

9138 

9143 

9149 

9154 

9159 

9165 

9170 

9175 

9180 

9186 

83 

9191 

9196 

9201 

9206 

9212 

9217 

9222 

9227 

9232 

9238 

84 

9243 

9248 

9253 

9258 

9263 

9269 

9274 

9279 

9284 

9289 

85 

9294 

9299 

9304 

9309 

9315 

9320 

9325 

9330 

9335 

9340 

86 

9345 

9350 

9355 

9360 

9365 

9370 

9375 

9380 

9385 

9390 

87 

9395 

9400 

9405 

9410 

9415 

9420 

9425 

9430 

9435 

9440 

88 

9445 

9450 

9455 

9460 

9465 

9469 

9474 

9479 

9484 

9489 

89 

9494 

9499 

9504 

9509 

9513 

9518 

9523 

9528 

9533 

9538 

90 

9542 

9547 

9552 

9557 

9562 

9566 

9571 

9576 

95^J 

9586 

91 

9590 

9595 

9600 

9605 

9609 

9614 

9519 

9624 

9628 

9633 

92 

9638 

9643 

9647 

9652 

9657 

9661 

9666 

9671 

9675 

9680 

93 

9685 

9689 

9694 

9699 

9703 

9708 

9713 

9717 

9722 

9727 

94 

9731 

9736 

9741 

9745 

9750 

9754 

9759 

9763 

9768 

9773 

95 

9777 

9782 

9786 

9791 

9795 

9800 

9805 

9809 

9814 

9818 

96 

9823 

9827 

9832 

9836 

9841 

9845 

9850 

9854 

9859 

9863 

97 

9868 

9872 

9877 

9881 

9886 

9890 

9894 

9899 

9903 

9908 

98 

9912 

9917 

9921 

9926 

9930 

9934 

9939 

9943 

9948 

9952 

99 

9956 

9961 

9965 

9969 

9974 

9978 

9983 

9987 

9991 

9996 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

2m 


530  ADVANCED  ALGEBRA 

In  finding  the  logarithm  of  64345  we  assumed  that  the  dif- 
ference of  two  logarithms  was  proportional  to  the  difference 
between  the  corresponding  numbers,  an  assumption  which  gives 
approximately  correct  results  if  the  difference  between  the 
numbers  is  small. 

Note.  For  greater  accuracy,  tables  of  more  places  should  be  used. 
Five-place  tables  usually  contain  the  mantissas  of  all  numbers  from  1  to 
9999,  and  are  constructed  in  the  same  manner  as  four-place  tables,  the 
only  difference  being  that  the  first  three  figures  are  found  in  the  column 
headed  JV,  while  the  fourth  one  is  at  the  top  of  the  other  columns. 

If  possible,  the  student  should  solve  all  examples  of  this  chapter  by 
means  of  a  five-place  table. 

Ex.  1.   Find  log  2762400. 
The  mantissa  of  276  =    .4409. 

Tabular  difference  =  16  ;  .24  x  16  =  4  (nearly). 

Hence  log  2762400  =  6. 4413. 

1414  .  27' 


Ex.  2.    Eind  the  log 


Let 


.072  .  ^102 

1414  .  27« 


.072  .  ^I02 

Then       log  x  =  log  1414  +  5  log  27  -  (log  .072  +  I  log  102) . 

log  1414  =    3.1504  log  .072  =  8.8573  -  10 

5  log  27  =    7.1570  I  log  102  =    .6695 

10.3074  9.6268  -  10 

9.5268  -  10 
logx  =  10.7806 

EXERCISE 

Verify  the  following  statements  : 

1.  log  44  =  1.6435.  4.   log  47210  =  4.6740. 

2.  log  .002  =  7.30103  -  10.  5.    log  14.566  ^  1.1634. 

3.  log  .791  =  9.8982 -10.  6.    log  1.8684  =  0.2714. 


APPENDIX 


531 


Find  the  logarithms  of  the  following  numbers : 


7.    154. 

13. 

4.2591. 

19. 

70951. 

24. 

506861. 

8.    2.34. 

14. 

.5271. 

20.    i 

84.827. 

25. 

21.1447. 

9.    .0456. 

15. 

.06217. 

21. 

.00035995. 

26. 

.075907. 

10.    67100. 

16. 

.002457. 

22.    : 

25.288. 

27. 

.0052208. 

11.   45.6. 

17. 

45.72. 

23. 

1.44598. 

28. 

10134700 

12.    1623. 

18. 

18712. 

Find  the 

logarithms  of  the 

following : 

29.  93x3514. 

30.  1225x387. 

42. 

47  X  .653 
3576  X 

X  12.83 
1520 

31.    628x493. 

43. 

.765  X  .0018 
31457  X  567.42 

32.    3748  X 

1752  X  4065. 

44. 

5^. 

33.   |. 

45. 

16». 

34.    ^, 

46. 

«)". 

35.    -1^. 

47. 

inr- 

36.    15f. 

48. 

(vr- 

37.    7/^. 

49. 

.5936^  X  < 
.07623 

386 

38.  f 

39.  ^1^  X  - 

138 

\r.          213    X    ' 

Tf  tf 

50. 

V5. 

f65 

r.655 
:718 

X  2.685 
X5944 

51. 
52. 
53. 
54. 
65. 

V73567. 
^135. 

40. 

3145  X 

^15276. 

5.5347 
•    137.65 

^35107. 

^^13. 

532  ADVANCED  ALGEBRA 

17.   To  find  a  number  whose  logarithm  is  given. 
Ex.1.    To  find  a;,  if  log  ic  =  7.1931  - 10. 

Find  1931  in  the  table  of  mantissas.  On  the  same  line  in 
the  column  headed  ^we  find  15,  the  first  two  figures  of  x/and 
at  the  head  of  the  column  which  contains  1931  is  6.  Hence  1, 
5,  6,  are  the  figures  of  x.  Since  the  characteristic  is  7  —  10,  or 
—  3,  a?  is  a  fraction,  and  two  ciphers  immediately  follow  the 
decimal  point. 

Whence  x  =  .00156. 

Ex.  2.    Given  log  a?  =  1.0476,  required  the  value  of  x. 

We  find  in  the  table  the  two  mantissas,  .0453  and  .0492,  cor- 
responding with  the  numbers  111  and  112. 

But  .0492  -  .0453  =  .0039, 

and  .0476  -  .0453  =  .0023. 

Hence  if  the  mantissa  .0453  increases  .0039,  then  x  in- 
creases 1. 

Therefore,  if  the  mantissa  .0453  increases  .0023,  then  x  in- 
creases If. 

Neglecting  the  decimal  point,  we  have 

0^  =  11111  =  11159. 
But  since  the  characteristic  is  1,  we  have  x  =  11.159. 

Ex.  3.    Given  log  x  =  8.4569  — 10,  required  the  value  of  x. 

The  next  lower  mantissa  in  the  table  is  4564,  corresponding 
with  the  number  286.  The  difference  of  the  two  mantissas  is 
5,  the  tabular  difference  is  15,  and  j\  =  .33. 

Hence  the  figures  of  x  are  28633. 

Considering  that  the  characteristic  is  —  2,  we  have 

i»  =  . 028633. 


APPENDIX  633 

EXERCISE 

Find  the  numbers  corresponding  to  the  following  logarithms : 

1.  4.7796.  5.   .0170.  9.    .78134. 

2.  .9430.  6.   .1038-3.  10.   5.61658. 

3.  .8949.  7.   1.07426.  11.   9.57938-10. 

4.  8.8376-10.         8.   3.59478.  12.    7.83145-10. 

13.  8.47652-10.     15.    How  many  digits  are  in  2^? 

14.  3.24567.  16.   Find  the  number  of  digits  in  2i« .  320 .  430. 

18.  Computation  by  Logarithms.  —  The  approximate  value  of 
an  arithmetical  expression  involving  only  multiplication,  divi- 
sion, involution,  and  evolution  may  be  found  by  logarithms. 

If  a  greater  logarithm  has  to  be  subtracted  from  a  less,  add 
to  the  minuend  10  - 10,  e.g.  for  2.4713  write  12.4713  -  10. 

Ex.  1.   Find  the  value  of  x,  if 


^     4.729  X 
62.7  X 

3.214 

8.392 

log  X  =  log  4.729  +  log  3.214 

-(log  62.7+ log  8.392). 

log  4.729=     .6748 
log  3.214=     .5071 

log  62.7  =  1.7973 
log  8.392=    .9239 

11.1819-10 
2.7212 

2.7212 

log  X  =  8.4607  -  10 

»  =  . 02889. 

Hence 

Ex.  2.   Find  the  value  of  a;,  if  a;  =  .872^ 
log  a;  =  7  X  log  .872 
=  7  X  9.9405  -  10 
=  69.5835-70 
=   0.5835-1. 

Therefore  x=     .38327. 


534  ADVANCED  ALGEBRA 

19.  To  divide  a  logarithm  whose  characteristic  is  negative, 
write  it  in  such  a  form  that  the  negative  portion  of  the  charac- 
teristic is  exactly  divisible  by  the  divisor.  Thus,  to  divide 
.4765  -  2  by  3,  write  it  in  the  form  1.4765  -  3 ;  to  divide 
8.4762  — 10  by  7,  write  it  in  the  form  12.4762  — 14,  or  more 
conveniently  68.4762  -  70. 

Ex.  3.   Find  the  value  of  x,  if 

05  =  ^1234. 

\ogx  =  j\\ogA2S4: 

=  ^2 .  (9.0913  -  10) 

=  3^2(119.0913-120) 

=  9.9243  - 10. 

Hence  a;  =  .84. 

-  .0239  X  ^iTS  X  .2' 


Ex.  4.   Eind  the  value  of 


42.3^ 


As  negative  numbers  have  no  real  logarithms,  we  determine  first  the 
value  of  the  right  member  without  regard  to  its  sign. 

I.e.  let 


^  _  .0239  X 

:  ^41.3  X  .27 
42.32 

a;  =  log  .0239  +  1  log  41.3 +  7 

'log.2-2: 

log  42.3. 

log  .0239 
i  log  41.3 
7  log  .2 

=  8.3784  - 10 

=  K1.6160) 

=  7(9.3010-10) 

=    8.3784- 
=     .5387 
=    6.1070- 

-10 
-10 

2  log  42.3 

=  2(1.6263) 

14.0241 . 
=   3.2526 

-20 

log  x  =  10.7715 -20 
X r= .0000000005909. 
Hence  the  required  value  =  -  .0000000005909. 


APPENDIX  535 

20.  The  cologarithm  of  a  number  is  equal  to  the  negative 
value  of  the  logarithm  of  that  number.  It  is  most  conven- 
iently obtained  by  subtracting  the  logarithm  from  0,  or  from 
10  - 10. 

E.g.  colog2=   0  — log2 

But  0  =  10  - 10 

log  2=      .30103 

colog2=   9.69897  -10 

Hence,  the  cologarithm  is  obtained  by  subtracting  every  sig- 
nificant figure  from  9  except  the  last,  which  is  subtracted  from 
10,  and  annexing  the  negative  characteristic  -  10. 

Note.     Since  log  -  =  log  1  —  log  x  =  0  —  log  a;,   the  cologarithm  of  a 

X 

number  is  also  the  logarithm  of  the  reciprocal  of  the  number. 

21.  The  addition  of  a  cologarithm  is  equivalent  to  the  sub- 
traction of  a  logarithm.  Hence,  examples  may  sometimes  be 
simplified  by  the  use  of  cologarithms.  As  a  rule,  however,  the 
gain  is  very  slight,  and  the  beginner  is  advised  not  to  use  co- 
logarithms  until  he  is  perfectly  familiar  with  logarithms. 

Ex.  5.    Find  the  value  of  x,  if 

^^47.32x41.92 
512.75 

log  X  =  log  47.32  4-  log  41.92  +  colog  512.75 

log  47.32  =  1.6751 
log  41.92  =  1.6224 
log  512.75  =  2.7099 ;  colog  512.75  =  7.2901  - 10 

logo;  =    .5876 

X  =  3.869 


536 


ADVANCED  ALGEBRA 


EXERCISE 

Find  the  value  of  the  following  expressions :  * 

1.  492.4x72.61.  4.    8.927x794.25. 

2.  6.417  X  .00342.  5.    6249  x  .003217  x  .412. 

3.  5.921  X  .072135.  6.    37.21  x  4729  x  .41115. 

7.  4.1929  X  (-4.789)  x  672. 

8.  .00423  X  (-  .0472)  x  (-  42196). 


9.    4719.1 -49.32. 
10.    .0048285 --(-.08235). 
6721  X  4.238 


11. 


12. 


13. 


14. 


19.425 

768.25  X  .49235 
.04216 

421.75  X  .06255 


15. 


16 


17. 


8.759236 
-.0576438* 

.000798543 
.000000965438* 

456.22  X  72.555  xl2« 


103^  X  497  X  .0456789' 

18.  1.357241i«. 

19.  1.26677^5. 


53.29  X  1.9985 

49876  X. 037542x68.7075    ^^'    -87^0581 
7.81649  X  578.93  x  28.4299 '   21.    8095.371-3. 

22.    4  Trr",  if  ,r  =  3.14159  and  r  =  2.0667. 


23.  214204^1. 

24.  39.679'A 


25. 


/3390  X  4.3401\" 


29.  ^567348. 

30.  -v/235.78. 

31.  a/33866. 


V      13814.4 

26.  .098756^. 

27.  \/8. 

28.  \/35246. 


32.  V1350|. 

33.  (317.75) «. 

34.  (i.6^)-«. 

35.  2.nS28^-^'l 
*  If  possible,  employ  a  five-place  table. 


APPENDIX  637 

36.     'VW^.  39.    ^lOOO-^lOO. 

„_       99.1767^  X  12.34  40 

" •  •    ,^^  „^^ — ^  „  . _.  „• 


(20.358  X  10.1575)3  V7 ^7 


52072  xV.00734^  lo^^ 

.2556088  •    ^f:^* 

22.  An  exponential  equation  is  an  equation  in  which  the 
unknown  quantity  occurs  as  an  exponent,  as  2*  =  7.  Such 
equations  are  readily  solved  by  the  use  of  logarithms. 

Ex.  1.    Find  the  value  of  x,  if 

23'=  =  923. 
Taking  the  logarithms  of  both  members, 
« log  23  =  log  923. 

Hence  x  =  ^^^^  =  ^^^  =  2.ns. 

log  23       1.3617 

23.  Change  of  system.  Logarithms  to  any  base  may  be 
readily  found  by  the  preceding  paragraph. 

Ex.  2.    Find  log;  12. 

Let  X  =  log7 12. 

According  to  the  principle  of  §  2,  we  have 

7^-  =  12. 

Taking  the  logarithms  of  both  members,  and  dividing, 

^^  log  12 
log  7 

Ex.  3.    Express  log^n  by  means  of  common  logarithms. 

Let  X  =  logft  n ; 

then  6*  =  n. 

Taking  the  logarithm  of  both  members,  and  dividing, 

3.  _  log  n 
log  b 


638  ADVANCED  ALGEBBA 

EXERCISE 
Solve  the  following  equations : 

1.  4^  =  18.                   5.    (.5)*  =  .10.  9.  -s/i0  =  2. 

2.  5*  =  16.                   6.    10'=+^  =  2'.  10.  a/145  =  3. 

3.  14^^  =  224.               7.    4^.5^  =  1700.  11.  ^1000  =  5. 

4.  7*  =  370.                 8.   4^=+! .  5^-1=  100.  12.  -\/20  =  10'. 

Find  to  three  places  of  decimals : 

13.  logy 8.  15.   log4  3.  17.    log4l9. 

14.  log^a  16.    logs  77.  18.    logy  66. 

19.    If  =  c,  find  (a)  the  value  of  x  in  terms  of  a,  b, 

5  —  1 

and  c ;    (b)  the  numerical  value  of  ic,  if  a  =  5,  6  =  10,  and 

c  =  ll. 

.  20.    The  nth  term  of  a  geometric  series  is  I,  the  first  term 
is  a,  and  the  common  ratio  is  r.     Find  n  in  terms  of  a,  I,  and  r. 

21.    Prove  that  logj  a  •  log„  6  =  1. 

VIII.    COMPOUND  INTEREST   AND  ANNUITIES 

1.   To  find  the  amount  a„  of  a  principal  of  p  dollars  for  n  years 
at  /?%  compound  interest. 

Then  ai=p(l+r), 


a„  =  p{l+rY.  (1) 

Taking  the  logarithm  of  both  members,  we  obtain 

log  a„  =  log  2?  +  n  log  (1  +  r).  (2) 


APPENDIX  539 

Equations  (1)  and  (2)  may  be  used  to  find  any  of  the  four 
quantities  a„,  jh  r,  and  n,  if  the  other  three  are  given. 

Ex.  1.    Eind  the  amount  of  $  1200  for  5  years  at  4%  com- 
pound interest. 

Substituting  the  given  values  in  (2), 

log  as  =  log  1200  +  5  log  1.04 

=  3.07918  +  5  X  .01703  =  3.16433. 
Hence  «6  ==  $  1459.93. 

Ex.  2.    In  how  many  years  will  $  100  amount  to  $  1100  at 
6  %  compound  interest  ? 
Substituting  in  (2), 

log  1100  =  log  100  +  n  log  1.05. 

Hence  „^  log  1100 -2 

log  1.05 

=  l^il^  =  49.1  years. 
.02119  ^ 

Note.     Since  formula  (I)  is  only  true  for  integral  values  of  w,  the 
fractional  part  of  the  preceding  answer  is  only  an  approximation. 

2.  The  compound  interest  /  is  the  difference  between  amount 

and  principal,  or :  #        /i    ,     n„  /o\ 

^         ^    '  /  =  p(l-\-ry-p.  (3) 

3.  If  the  interest  is  compounded  semiannually,  the  amount  aj 
is  obtained  by  the  method  of  §  1. 

..-,(i+0--  (.) 

4.  An  annuity  is  a  fixed  sum  of  money,  payable  at  equal 
intervals  of  time. 

5.  To  find  the  amount  A„  of  an  annuity  of  5  dollars  left  unpaid 
for  n  years  at  /?  %  compound  interest. 


540  ADVANCED  ALGEBRA 

The  s  dollars  due  at  the  end  of  the  first  year  will  amount  in 
the  remaining  n  —  1  years  to  s{l  +  r)"~^  dollars. .  Similarly  the 
payment  due  at  the  end  of  the  second  year  will  at  the  end  of 
the  entire  period  amount  to  s(l  +  r)"~-  dollars,  etc. 

Hence     '    A  =  s(l  +  >*)"~^  +  s(l  +  ry-^  + h  s. 

Adding  the  G.  P.  (§  365) 

An  =  '-[.(l+ry-l\  (5) 

Again,  this  formula  may  be  used  to  find  any  of  the  four  quan- 
tities A,  s,  r,  and  n,  if  the  other  three  are  known. 

6.  To  find  the  present  value  Aq  of  an  annuity  of  s  dollars  to 
continue  for  ?i  years  at  jR%  compound  interest. 

At  the  end  of  n  years  ^  would  amount  to  ^(1  +  r)^ 

The  annuity  at  the  same  time  would  amount  to  -  [(1+^)''— 1]. 


r 


Hence  A(l  +  ry  =  -  [(1  +  r)"- 1]. 

r 

Therefore  Ao  =  -  [l-ji^ — ^1- 


(6) 


Ex.  3.    Find  the  present  value  of  an  annuity  of  $  800,  for  20 
years,  allowing  compound  interest  at  4  %  per  annum. 

A  _800r.        1    1 

log  (^— i— ^  =0-20  log  1.04 

=  0  -  .3406  =  .6594  -  1. 
1 


(1.04) 


20 


.45644. 


Hence  Ao  =  i^L^  =  $  10871.20. 

.04 


APPENDIX  641 


EXERCISE 


Find  the  amount  at  compound  interest 

1.  Of  $  200  for  8  years,  at  ^  %. 

2.  Of  $  1550  for  5  years,  at  5  %. 

3.  Of  f  1  for  1000  years,  at  4  %. 

4.  Of  $  20  for  20  years,  at  2  %,  interest  compounded  semi- 
annually. 

5.  Find  the  compound  interest  of  %  250  for  10  years  at  6  % 
compound  interest. 

6.  Find  the  principal  that  will  amount  to  ^  1000  in  20 
years  at  4^  %  compound  interest. 

7.  Find  the  principal  that  will  amount  to  $420  in  10  years 
at  5  %  compound  interest. 

8.  In  what  time  will  $8007  amount  to  $21218  at  ^% 
compound  interest  ? 

9.  In  what  time  will  $1  amount  to  $5000000  at  5/^% 
compound  interest  ? 

10.  At  what  rate  will  $  200  in  10  years  amount  to  $  350  at 
compound  interest  ? 

11.  At  what  rate  will  $40  in  12  years  amount  to  $80, 
interest  compounded  semiannually  ? 

12.  Find  the  value  of  an  annuity  of  $  40  left  unpaid  for  10 
years,  5  %  compound  interest  being  allowed. 

13.  Find  the  present  value  of  an  annuity  of  $800  for  10 
years,  4  %  compound  interest  being  allowed. 

14.  Find  the  present  value  of  an  annuity  of  $500  for  4 
years,  5  %  compound  interest  being  allowed. 

15.  Find  the  present  value  of  an  annuity  of  $900  for  12 
years,  5  %  compound  interest  being  allowed. 

16.  What  annuity  can  be  purchased  for  $  2500,  if  it  is  to  run 
for  10  years,  and  6  %  annual  compound  interest  is  allowed  ? 


542  ADVANCED  ALGEBRA 


IX.    SUMMATION  OF   SERIES 

1.   The  sum  of  /(I)  +/(2)  +/(3)  •-.  +/(n)  is  frequently  de- 

n  n 

noted  by  %xf(x)  or  %f(x)* 

Thus,  12  +  22  +  32  +  ...  +  w2  =  s  (x2). 

Ill  1     «1 

12     3       .       n      ix 


Vl  +  1  +  Vl  +  3  +  Vl  +  3  +  ...  +  Vl  +  w  =  S  Vl  +  x. 

8 

log  5  +  log  6  +  log  7  +  log  8  =  2  log  x. 

5 

2.  If  the  lower  limit  is  omitted,  1  is  understood,  and  if  the 
upper  limit  is  omitted,  ?i  is  understood. 

Thus,  2  (a;3)  =  2  (x^)  =  13  +  23  +  3^  +  43. 

1 

2/(a^)=/(l)+/(2)+/(3)  +  -  +/(n). 
2(1  +  ay  =  (1  +  a)  +  (1  +  a)2  +  (1  +  a)^  +  ...  +  (1  +  a)«. 

3.  S  [f(x)  +  F(;r)]  =  Sf  (x)  +  ^F(x).  (1) 
For,  S[/(a^)  +  F{x)]  =  (/(I)  +  F(l))  +  (/(2)  +  F(2)) 

+  (/(3)  +  F(3))  +  ...  +  (f(n)  +  F{n)) 
='[/(!) +/(2)+/(3)H--+/(n)] 
+  IF(1)  +  2^(2)  +  i^(3)  +  ...  +  F(n)-] 
■        =%f(x)  +  %F(x). 
Thus,  S(a;2  +  x)  =  2  (x2)  +  2  (x). 

2 (3  x2  +  2  X  -  1^  =  2(3  x2)  +  2(2  X)  -  2 (ly 

*  2  is  the  Greek  letter  Sigma. 


APPENDIX  643 

4.  If  m  denotes  a  constant,  then 

S/n^(jr)=/nS/(jr).  (2) 

For,  ^mf(x)  =  m/(l)  +  m/(2)  +  m/(3)  +  -  +  mf(n) 

=  m[/(l)  +/(2)  +/(3)  +  -  +/W] 

=  m2/(a;). 
Thus,  S(3x4)=3S(x*). 

2(3  X*  +  2  x3)  =2(3  a;*)  +  2(2  x^)  =  3  2(a;*)  +  2  2(a;3). 

2(73  a;)  =  73  2(x)  =  73(1  +  2  +  3  +  4)  =  730. 

5.  If  the  quantity  that  follows  the  summation  symbol  (2) 
does  not  contain  a  variable,  all  terms  of  the  sum  are  equal. 

Thus,  2(1)=  1  +  1  +  1  +  •••  +  1  =n. 

2(5)  =5  +  5  +  64-  •••  to  n  terms  =  5  n. 

Ex.  1.  Write  the  series  which  is  represented  by  2(«+l)(a;-|-5). 
2(x  +  l)(x  +  5)  =  2.6  +  3-7  +4-8+  ...  +(n  +  l)(w  +  5). 

Ex.  2.    Write  the  following  series  in  the  abbreviated  form : 

1.3-5  +  2.4.64-3.5.7  +  ...  +ri(n  +  2)(ri  +  4). 
Evidently  the  series  =  2x(x  +  2)(x  +  4). 

4 

Ex.  3.    Find  the  numerical  value  of  2  (x  •  2*). 

S(x  .  2*)  =  1  .  2  +  2  .  22  +  3  .  28  +  4  .  2* 
=  2  +  8  +  24  +  64  =  98. 

Ex.  4.    Simplify  2  (2  a^  +  3)1 

2(2  x2  +  3)2  =  2(4  X*  +  12  x2  +  9) 

=  2(4x4)+ 2(12x2)+ 2  9 
=  4  2(x*)+12  2(x2)+9n. 


544  ADVANCED  ALGEBRA 

Ex.  5.    If  ^f(x)  =  12,  and  %F{x)  =  2i,  find  %l3f(x)-5F(x)]. 
S[3/(x)-  5i^(x)]  =  S(3/(x))-  ^i5F(x)) 
=  3  S/(x)  -  5  2i?'(x) 
=  3  .  12  -  5  .  f  =  23^, 

EXERCISE 

Write  the  series  which  are  represented  by  the  following 
symbols : 


1. 

sL 

8. 

:Sa=. 

X 

9. 

:^x  .  a^ 

2. 

^{x  +  l)(x  +  2){x'{-3y 

3. 

i(iogxy 

10. 

:S(2a;4-lK 

4. 

%(af). 

11. 

SC^V). 

5. 

x^+1}^ 

12. 

S*a» 

x" 

4 

13, 

SU- 

6. 

%{a-]-x). 

7. 

.^u+^\ 

14. 

V°0,.a^ 

Write  the  following  series  m  the  abbreviated  form : 

15.  1.2  +  2.3  +  3.4+ ••• +n(w4-l). 

16.  1  .  22  +  2  .  32  +  3  .  42  +  ...  +  n(n  +  ly. 

1^     2^3     3-4  n(n  +  l) 

3. 4*^4. 5^5. 6^      ^(n  +  2)(n  +  3) 

18.  2  +  4  +  6  +  8  +  .».  +271. 

19.  1+3  +  5  +  7 +... +(2  n-1). 

20.  1.2.3  +  2.4.5  +  3.6.7  +  ...+w(27i)(2n  +  l; 

21.  1  .3  +  2.5  +  3.7  +  4  .9  +  ...to  w  terms. 

22.  1.2.1  +  2.4.3  +  3.6.5  +  4.8-7  +  ...ton  term^ 


APPENDIX  545 

23.  J  +  i  +  i  +  ••  •  to  n  terms. 

24.  l  +  2a  +  3a^  +  4a^  4- •••  to  ?i  terms. 

100     100.99     100-99.98  ,^,,,^, 

"^         1    ^    1.2    ^  [3 

Find  the  numerical  value  of : 

26.    ^i.  29.    ^(a^  +  l)(a^  +  2). 


0? 
5 

27.    2i»2. 


30.    ^{x^-\-x). 


^      1 
31.    S 


28.    :S(7).  «  +  2 

Simplify : 

32.  2  (4  a; +  2).  36.    Sa;(i»  +  1)'. 

33.  :S(9a?-7).  e 

34.  2(2c^  +  3a:  +  5).  ^7.    2[/W -/(..- 1)]. 

35.  ^ (^3  0;^  + 5  0^-^1  38.    S[/(a^) -/(o^-l)]. 

If:S(^)=^(^  +  ^\  and  S(^^  =  ^^^  +  ^^f^  +  ^\  find: 

39.  S(2iB4-l).  43.  :S (a; 4-1) (a? +  2). 

40.  :S(3ir-2).  44.  :S(a;  +  4)l 

41.  S(a^-2a;).      .  45.  1.3  +  2.5  +  3.7  +  ..' 

42.  ^{2x'-4.x-\r^)'  +w(27i  +  l). 

Prove  that : 

46.  lf(x)-^f{x)=f{n). 

47.  2[/(a;  +  l)-/(aj)]=/(n  +  l)-/(l). 

48.  %lf{x  + 1)  -/(a^  - 1)]  =/(n  + 1)  +/(n)  -/(I)  -/(O). 

2n 


546  ADVANCED  ALGEBRA 

6.    S  [f(x)  -  f(x  - 1)]  =  f(n)  -f(0).  (3) 

For,         2[/(^')-/(«^-l)]=/(l)-/(0) 

+/(3)-/(2) 


+/0z-l)-/(n-2) 

But  this  obviously  equals  f(n)  -  /(O). 

7.    Formula  (3)  can  be  used  to  determine  the  sum  of  a  num- 
ber of  series. 

Ex.  1.    Let  f(x)=x\ 
Substituting  in  (3),  we  obtain 

2[X2-(X-I)2]=w2. 

Simplifying,  2(2  x  -  1)=  n^. 

■  Hence  2(x)  =  ^' +  ^  =  !?l^_±i). 

^  ^  2  2 

I.e.  1  +  2  +  3  +...  +  »  =  ViilL+lL. 

Ex.  2.    Let  /(i»)  =  a^. 
Substituting  in  (3),  we  obtain 

2[x8-(a;-l)3]=n«. 
Simplifying,  2(3  a;2  -  3  x  +  1)  =  w*. 

Or,  3  2(ic2)  -  3  2(x)  +  n  =  n8. 

Substituting  the  value  of  2(cc),  (Ex.  1),  and  transposing, 

32(a;2)=ng  +  ^^(^  +  ^)-n. 


2 

c 

6  6 


2ra;2)  =  2  w3  4-  3  ^2  4-  n  _  n(n -\-  l)(2n  +  l) 


APPENDIX  547 


6 
E.g.  12  4.  22  +  32  + ...  +  802  =  ^Q  •  ^1  •  ^^^  =  173880. 


8.  In  a  similar  manner  the  sum  of  the  first  n  natural  num- 
bers raised  to  any  power  may  be  found,  provided  the  sums  of 
the  lower  powers  are  known. 

To  find  ^(a:^)  make /(a;) =3?'*,  to  find  %(x^)  make/(a;)=.'c^,  etc. 

9.  The  sum  2/(ic),  iif{x)  is  a  rational  integral  function,  can 
be  reduced  to  the  sum  of  the  following  series : 

S(x)  =  ri(!^,  (4) 

S(^)  ="'("  + l)%tc.  (6) 

Ex.  3.   rind  the  value  of  %(l2x'-ix  +  3). 

6  2 

=  4  w3  +  4  n2  +  3  n. 

Ex.  4.   Find  the  sum  of  the  series 

1.2.3  +  2.3.4  +  3.4.5+  ...  to  w  terms. 
The  series  =  ^x(x  +  1)  (x  +  2) 
=  S(x8  +  3  a;2  +  2  «) 
=  2(x8)+3  2(x2)+2S(a;) 

^n2(n  + 1)2^3.  n(n  +  l)C2n  +  l)^^(>^^l^ 
4  6 

4 

=  in(w  +  l)(n  +  2)(n  +  3). 


648  ADVANCED  ALGEBRA 


1.   Prove  that  :S(a;^)  = 


EXBBOISB 
-n(7i+l)T. 


.  T 


2.  Prove  that  5 (a:*)  =  >K^  +  l)(2n  +  l)(3^^4-3n-l)^ 

•      ^    ^  2.3.5 

Find  the  sum  of  the  following  series : 

3.  13  +  23+33...  +  100^  6.    2(6a;2  +  4a;  +  l). 
10 

4.  ^{x").  7.    :S(3a^  +  2x-l). 

5.  2(3a;-l).  8.    %{x^l){^x  +  2). 

9.  1  .2  +  2.3  +  3  .4+...  to  w  terms. 

10.  1  .3  +  2.4  +  3  .5  +  ...  ton  terms. 

11.  3  .  7  +  4.9  +  5  .11  +  ...  to  w  terms. 

12.  1.22  +  2.32  +  3.42+  ...+n(n  +  l)2. 

13.  1.2.2  +  2.3.4  +  3.4.6  +  ...  +w(n  + 1)2  n, 

14.  1.2.3.4  +  2.3.4.5  +  3.4.5.6  ...  to  71  terms. 

15.  Find  the  sum  of  the  first  n  odd  numbers. 

16.  Find  the  sum  of  the  first  n  even  numbers. 

17.  Find  112  +  122  +  132+  ...to  40  terms. 

100 

18.  Find  2(a5^). 


10.   Arithmetic  series.     Let 


\k 
Then  -  /(a:  - 1)  =  ""^C^, 

and  f{x)  -fix  - 1)  =  *-^C»_i  (§  381). 


APPENDIX  549 

Substituting  in  the  general  formula  (3), 

By  substituting  for  k  respectively  the  values  2,  3,  4,  we 
obtain 

S'-^Ci  =  :S(aj-l)=^?^^^;=^,  (7) 

Le.     0  +  1+2 +  ...+(^-1)  =  ''^^^^^ 

0  +  0  +  l  +  3  +  6+...=^(^^-^)/^-^),  etc. 

11.  If,  in  a  series,  each  term  is  subtracted  from  the  follow- 
ing one,  we  obtain  the  series  of  first  differences.  If  each  term 
of  the  series  of  first  differences  is  subtracted  from  the  follow- 
ing one,  we  obtain  the  series  of  the  second  differences,  etc. 

Thus,  consider  the  series     1        3        9        22        45        81 

First  differences,  2        6       13        23        36 

Second  differences,  4        7        10        13 

Third  differences,  3        3  3- 

Fourth  differences,  0         0 

12.  An  arithmetic  series  of  the  /ith  order  is  a  series  whose 
>«th  differences  are  all  equal. 

Thus,  an  ordinary  arithmetic  progression  may  be  considered  an  arith- 
metic series  of  the  first  order.  The  series  considered  in  §  11  is  an 
arithmetic  series  of  the  third  order,  etc. 


650  ADVANCED  ALGEBRA 

13.  If  we  consider  the  following  series  and  its  first  differ- 
ences, 

Series,  «!        Og        a^       •••       a^ 

First  differences,  61         62         K-i 

it  is  obvious  that 

«x  =  «i  +  (&i  +  &2  +  •  •  •  +  K-i)'  (10) 

14.  In  an  arithmetic  series  of  the  first  order  the  first  differ- 
ences are  all  equal,  hence 

If  we  denote  the  sum  of  the  first  n  terms  of  a  series  by  s„, 

we  obtain  «       -« /^  \ 

s«  =  2(a^).       .  ■      , 

Hence,  for  an  arithmetic  progression,  we  have 

5„=:S[ai  +  (aj-l)6], 

s„  =  nai  +  -\- — ^  61.  (11) 

Note.  These  results  were  obtained  in  slightly  different  form  in 
Chapter  XXI. 

15.  Let  an  arithmetic  series  of  the  second  order  and  its 
two  series  of  differences  be  represented  as  follows : 

Series,  Oi        ag        a^        •••         a. 

First  differences,  bi        bz        •••         6^_i 

Second  differences,  Cj        •••        Cj 

Then  a,  =  ai  +  (&i-f  52-&X-1).     (§13) 

But  bi  +  b2'-'bj,_i  is  an  arithmetic  seriep  of  the  first  order, 
which  may  be  added  by  formula  (11).  Hence,  making 
n  =  x  —  l,  we  have 

a,  =  a,  +  (x-l)b,  +  ^'-'^}^''-^\,. 


APPENDIX  551 

But  s„  =  2a^ 

Or,  considering  §  10, 

s„  =  nai  +    ^  ^     ^61  + -|^ ^Ci. 

16.   Similarly,  we  may  treat  the  arithmetic  series  of  the 
third  order. 

Series,  aj        ag        as        a^        a^        •••         a^ 

First  differences,  61         62         ^3         ^4         ••*         K-i 

Second  differences,  Ci         Cg         C3        •••         c,_2 

Third  differences,  d^        d^        •••         c?i 

a,  =  ai  +  (&i  +  fe2+--+&x-i)     (§13) 

=  tti  +  (a?  - 1)  61  +  ^ f^ ^  Ci 

if 

_^(^,l)(^-2)(a.-3)^^^     (§15) 
And      s„  =  Sa, 

_^^^(^-lK^-2i^-3i. 

^                          ,  n(n  — 1),     ,  w(7i  — l)(w  — 2)^ 
Or,         «n  =  wai  +  -^-77j — ^&i  +  -^ r^^^ ^Ci 

,  y,,rn-l)(7i-2)(n-3)^ 
+  g ^(^. 


562  ADVANCED  ALGEBRA 

17.  By  mathematical  induction  it  can  be  shown  that  these 
results  are  true  for  arithmetic  series  of  any  order.  Omitting 
the  subscript  1  of  the  first  term  of  each  series  of  differences, 
we  have  in  general : 

fl.  =  fli  +  (jr  -  1)6  +  ^^  ~  Vi^  -  ^^c  +  ».. 

Sn  =  noi  +  -^— — ^b  +  -^ ^ Lc  +  .... 

Ex.  Find  the  10th  term  and  the  sum  of  the  first  8  terms 
of  the  series  1,  3,  8,  20,  43,  81. 

Series,  1        3        8        20        43      81 

First  differences,  2        6        12        23      38 

Second  differences,  3        7        11        15 

Third  differences,  4         4         4 

Hence,  a  =  1,  6  =  2,  c  =  3,  (2  =  4. 

aio=l  +  9.2+^.3  +  fl|^.4  =  463. 

1  •  ^  1  •  ^  •  o 

,,  =  8.1  +  |^.2  +  i^.3  +  f-^I^.4  =  512. 


EXERCISE 

1.  Find  the  11th  term  of  3,  6,  11,  18,  27,  .... 

2.  Find  the  9th  term  of  2,  6,  12,  20,  30,  .... 

3.  Find  the  10th  term  of  2,  9,  28,  65,  126,  217,  .... 

4.  Find  the  7th  term  of  2,  3,  7,  15,  28,  47,  .... 

5.  Find  the  20th  term  of  1,  4,  10,  23,  47,  .... 

6.  Find  the   sum  of  the  series  1,  4,  10,  23,  47,  .••  to  10 

terms. 

7.  Find  the  sum  of  the  series  10,  28,  56,  94,  142,  200,  .••  to 
n  terms. 


APPENDIX  663 

8.  Find  the  sum  of  the  series  3,  23,  71,  159,  299,  503,  •••  to 

8  terms. 

9.  Find  the  sum  of  the  series  1, 1-f  2, 1+2+3,  1+2+3+4, 
•  ••  to  10  terras. 

10.  Find    the    sum    of   the   series   V,   1^  +  2^12  +  22  +  32, 
12  +  2^  +  32  +  42,  ...  to  9  terms. 

11.  Show  that  the  series  represented  by  2(2  aj2  _  4  a; -f  9)  is 
an  arithmetic  series  of  the  second  order. 

12.  Show  that  the  series  represented  by  '%{7?  —  ^x)  is  an 
arithmetic  series  of  the  third  order. 


18.  Many  fractional  series  can  be  added  by  means  of  formula 
(3).  Values  of  f{x)  which  contain  the  factor  x  in  the  denomi- 
nator, however,  should  be  avoided,  since  f{x  —  1)  would  be- 
come 00,  if  aj  =  l. 

Ex.  1.    Let /(ic)  =  -J: — 
•^^  ^      x  +  1 

Substituting  in  (3),         S  f  — V\  =  _! j. 

\x+l     xj     n  +  1 

Simplifying,  2 


x{x-\-\)      n  +  1 
Le.  1    +    1    +    1^  +  ...+        1  n 


1-2     2.8     3.4  n(n  +  l)      n  +  1 

19.   Similarly.        /(.)=^_^i_^ 

leads  to  the  value  of     2  — ^ 7-7 -r-y 

x{x-\-l)(x-^2y 

f(x)  = 

•'^  ^      (x  +  l)(x-i-2)(x  +  S) 

leads  to  2— -— -— ,    etc. 

a;(a;  +  l)(a;  +  2)(a;  +  3)' 


654  ADVANCED  ALGEBRA 

Ex.  2.    Find  the  sum  of  the  series 

~^77~S ~A ^~7^~^Ty -A ^       r.       rr-^ ^^    ^   teimS. 


1.2.3.4.5  2.3.4.5.6  3.4.5.6.7 

According  to  §  19,  let    /(x) 


(a;  +  l)(x  +  2)(x  +  3)(a;  +  4) 
Substituting  in  (3),  and  simplifying,  we  obtain 

1  1  1 


a;(a;+ l)(a;  +  2)(a;+3)(x  +  4)      96      4(w  +  l)(n+2)(w  +  3)(w  +  4) 
20.   To  add  a  series  of  the  form 

"^  7 — TIaT — ,  oT.\  +  7 7n^/ TTTT  +  •••  to  ^  termsj 


a(a  +  6)      {a-^b){a  +  2b)      (a  +  2  6)(a  +  3i)) 

let  /(aj)=_i 

-^^  ^     a  +  bx 

Thus,  to  find  -1-  +  -^  +  —^ to  n  terms,  let  /(ic)  •* 


2.5     5.8     8.11  '        ""  "     2  +  3a: 

Substituting  in  (3) , 

Ef-i ^V-^^ -• 

V2  +  3a:     3x-iy     2  +  3^1     2 

Hence,  v.  1  _         « 


(3a;- l)(3x  +  2)      2(2 +  3n) 
Le.  —  +  —  +  —^  +  ...  to  n  terms  =         ^ 


2.6      5.8     8.11  2(2  +  3  w) 

21.   The  series  - — 7  + ?r— p  +  tt^  +  T"^""  ^^  equal  to  three 
1.4     2.5     3.6     4.7 

series  of  the  preceding  kind,  viz. 

Vl-4^4.7     7.10     y     V2.5     5.8     8.11     J 

1,3.6      6.9      9.12     J 


or 


APPENDIX  555 

22.  An  infinite  series  'is  convergent  if  the  sura  of  its  first  n 
terms  approaches  a  finite  limit,  as  n  is  increased  indefinitely. 

The  limit  of  the  sum  thus  obtained  is  called  the  sum  of  the 
infinite  series. 

Thus,   «!  +  ^2  +  ^3  +  •••  +  <*n    01'  ^(^x)    IS    convergeut    if 

lim    2  (a^)  is  a  finite  number.      The  value  of  S  (ax)n= 

2(a^)  is  the  sum  of  the  infinite  series. 

E.g.  the  series 1 1 to  infinity,  is  convergent,  and  its 

1  •  2      2  •  o      o  •  4 

sum  equals  1,  for  lim  f-^^'\        =  1.   (§  18) 
\n  +  1/  „=« 

Similarlv  s =  —  i  ^^^  the  series  is  con- 

^'Tx(x  +  l)(x  +  2)(x  +  3)(x  +  4)      96 

vergent  (§  19). 

EXERCISE 

¥ind  the  sum  to  n  terms,  and  to  an  infinite  number  of  terms, 
of  the  following  series  : 

-     1    +   1^+1,+.... 


•    1.2-3     2.3.4     3.4.5 
2.   _l         +_!__+        1 


I02.3.4     2.3.4.5     3. 4»5.6 

1.3      3.5      5.7 

2.4^4.6     6.8^ 
111 

3-7^7.11     11-15 


556  ADVANCED  ALGEBRA 


8.    -J^+.     ,J    .„..+  1 


a(a-^b)      (a +  ?>)(«  4- 2  6)      (a  +  2&)(a4-3  6) 

9.    _J_  +  _l_  +  ^  +  .... 
2.3.4     3-4.5     4.5-6 

Hint.    Let/(a;)=       a^(^+l) 

10.  By  substituting /(a?)  =  ar*,  derive  the  formula  for  the 

sum  of  a  geometric  progression,  viz.  Sa?-^"^     (§  366) 

11.  By  substituting  f(x)  =x  >  r^,  derive  the  formula  : 

(r  - 1)2 

12.  Find  1  +  2  -  2  +  3  .  2^  +  4  .  2^  -h  •••  to  n  terms. 

13.  Find  1  +  2  •  5  +  3  .  5^  +  4  .  5^  +  ---  to  11  terms. 

14.  Find  l-r  +  2?-2  +  3r^  +  4r^  +  --- t0  7i  terms. 

15.  Find  1  -  2  -  2 +  3  -  2^-4  -  2^+ ...  to  n  terms. 

16.  Find  l-l  +  2-1.2  +  3-1.2.3..-  +  n|w. 
Hint.     Let  /(x)  =  |a;+  1. 

Derive  a  series  by  substituting  in  formula  (3). 

17.  f(^)  =  '^(3'^+i>  .      20.  m  = ^^^+A) 

22     f(x)=       '^•(^  +  ^)       - 
19     .^^.x(x  +  l)(x-^2)  ''^^      (x  +  ^){x  +  5) 

^      ^  [3  23.  f(x)  =  a^i^. 

Note.     Students  who    are    familiar  with  trigonometry  may  apply 
formula  (3)  to  trigonometric  functions. 


APPENDIX  557 

E.g.  by  substituting  f(x)  —  sin  (^  +  ^x),  we  obtain 

i-i?cos('^  +  -5'\ 


sin- 
Scos[^+(x- J)5]  = 


.    5 

sni  — 

B  ^ 

Or,  substituting  C for  ^, 

sin^^^cos^C  +  ^^-^^'j 


sin  — 

2 

sin -^  cos  ^^^B 
2               2 
^.gf.     cos  5  +  cos  2  5  +  cos  3  5  +  .••  +  cos  nB  — ■•- 

sin  — 
2 
Or, 

cos5  +  cos3  5  +  cos5B+  ...  +  cos(2w-  i)5-Sin2w5 


2sin^ 


IT    , 2  IT    , 3  IT    ,  , WIT 


Or,  cos-  +  cos^!-^  +  cos':^+ ... +cos^^  =  -l. 

n  n  n  n 

2               2 
sin  5  +  sin  2  5  +  sin  Z  B  +  ...  +  sin  w  5  = . 

sin^ 

2 

sin  5  +  sin  3  1?  +  sin  5  5  +  ...  +  sin  (2  n  -  1)5  =  51^1^. 

sin  5 

sin  31  +  sin  ^^^  +  sin  ^i^  + h  sin  '-^  =  cot 

n  n  n  n  2n 


INDEX 


[Numbers  refer  to  pages.] 


Abscissa 275 

Absolute  term 312 

"       value 5,  380 

Addition 19,  128,  248 

graphic     .     .     .     379,  382 

Aggregation,  signs  of  ....     12 

Algebraic  expression  .     .     .     .     12 

"        sum    ......     20 

Alternation 171 

Amplitude 380 

Annuities 539 

Antecedent    , 167 

Arithmetic  mean 338 

"  progression     .     .     .  334 

Arrangement  of  expressions      .     23 

Associative  law 37 

Axiom 65 

Axis  of  abscissas 275 

"    of  imaginaries     ....  379 
"    of  real  numbers  ....  379 

Base  of  a  logarithm     .     .     .     .619 

"    of  a  power 9 

Binomial 13 

Binomial  coefficients  .     .     .     .351 

"       theorem 348 

Biquadratics 496 

Brace 12 

Bracket 12 

Cardan's  formula 494 

Character  of  roots 327 

Checks       .     .     .     .     .     .  22,  43,  57 

Clearing  equations  of  fractions,  147 

Coefficient 10 

Co-factor 406 


Cologarithm 535 

Combination 394 

Commensurable  roots       .     431,  463 

Common  difference      ....  334 

"        multiple,  lowest,    118,  366 

"        ratio 340 

Commutative  law 37 

Complete  equations     ....  431 
Completing  the  square     .     .     .  292 

Complex  fraction 141 

"       numbers  .     .     .     370,  380 

Composition 171 

Conditional  equations      ...     64 
"  inequalities  .     .     .  360 

Conjugate  surds 254 

Consequent 167 

Consistent  equations   ....  185 

Constant 272 

Continuous  function    ....  458 

Convergent 555 

Coordinates 275 

Cross  product 49 

Cube  root 367 

Cube  roots  of  unity      ....  492 

Cubic  equations 492 

"     parabola 486 

Degree  of  a  term 43 

"       of  an  equation     .     278,  812 
Depression  of  an  equation    .     .  438 

Descartes'  Rule 452 

Descartes'  solution  of  biquad- 
ratics      496 

Detached  coefficient    ....  359 

Determinants 401 

"  elements  of     .     .401 


569 


560 


INDEX 


Determinants,  factoring  of 

Difference 

Discriminant       .... 
Discussion  of  problems     . 


420 

25 

327 

210 


Distributive  law 41 

Dividend 53 

Division 65 

Divisor 53 

Bliminant 426 

Elimination 185,  426 

Equations 64 


complete     .     .     .     .431 

consistent  .     .     .     .185 

cubic 492 

equivalent  ....  265 

exponential     .     .     ,  537 

fractional    ....  147 

graphic     representa- 

tion of     ....  278 

graphic  solution,  278,  284 

incomplete       .     .     .431 

indeterminate       .     .  512 

in  quadratic  form, 

278,  304 

linear     ...       65,  186 

literal     ...       65,  152 

numerical  ....     65 

quadratic    ....  289 

reciprocal  .     .     .     .497 

simple 65 

simultaneous. 

184,  312,  424 

" 

symmetric       .    312,  319 

Equivalent  equation    ....  265 

Evaluation  of  a  determinant     .  417 

Evolution 

218 

Expansion  of  a  determinant      .  402 

Exponent 

9 

Exponent 

s,  law  of  .     .     39,  63,  230 

Extraneous  roots 265 

Extreme 

169 

Factor    . 

06 

Factor  theorem    .     .     .     .267,  434 

Factor,  H.  C 114,  363 

Factoring  .  .  .96,  267,  331,  360 
Fourth  proportional  ....  169 
Fractional  equations  ....  147 
"  exponent  ....  231 
Fractions 122 

Geometric  progression     .     .     .  340 

Graphic  representation  of  cubic 
functions 488 

Graphic  solution  of  cubic  equa- 
tions       484 

Graphic  solution  of  equations, 

278,  470 

Graphic  solution  of  quadratic 
equations 308 

Graphic  solution  of  simultane- 
ous equations 284 

Graph  of  a  function     ....  274 

Grouping  terms 109 

Highest  common  factor  .     114,363 

Homogeneous  equations  .     312,  316 

"  polynomials  .     .     43 

Horner's  method 476 

Identical  equations      .     .       64,  156 
"        inequalities  ....  360 

Identities 64,  156 

Imaginary  numbers     .     .     218,  370 

"  roots 445 

"  unit 370 

Incommensurable  roots, 

431,  445,  470 
Incomplete  equations  ....  431 
Inconsistent  equations  .  .  .  185 
Independent  equations  .  .  .  185 
Indeterminate  equations  .     .     .  512 

Index 11 

Induction,  mathematical      .     .  346 

Inequalities 358 

Inferior  limit 456 

Infinite 366 

"       G.P 343 

Infinitesimal 366 


INDEX 


561 


Insertion  of  parentheses  . 
Integral  expression  .  . 
Interpretation  of  solutions 

Inversion 

Involution 

Irrational  numbers      .     . 


30 

96 

209 

171 

212 

242,  376 


Known  numbers 


Law  of  exponents  .     .     39,  53,  280 
Laws  of  signs     ...     37,  53,  212 

Like  terms 20 

Limit 365 

Linear  equation      ...       65,  184 
Literal  equations     .     .     .       65,  152 

Logarithm 519 

"        common       ....  520 
Lowest  common  multiple     118,  366 

Mantissa 525 

Mathematical  induction  .     .     .  346 

Mean,  arithmetic 338 

"      geometric 341 

Mean  proportional 169 

Member,  first  and  second     .     .     64 

Minor 405 

Minuend 25 

Modulus 380 

Monomials 12 

Multiple,  L.  C 118 

Multiple  roots 437 

Multiplicand 36 

Multiplication     ....       36,  250 
Multiplier 36 

Negative  exponents    ....  233 

"       numbers 4 

Newton's  method 464 

Order  of  a  determinant   .     .     .  402 

"      of  operations     .     .     .     .  13 

"      of  surds 242 

Ordinate 273 

Origin 275 

2o 


Parabola 486 

Parenthesis'* 12,  30 

Perfect  square 104 

Permanence 462 

Permutation 387 

Polynomial 12 

Polynomials,  addition  of      .     .    22 
"  square  of    ...    49 

Power 9 

Present  value  of  annuity      .     .  640 

Prime  factors 96 

Principal  diagonal 402 

Problem    .  1,  76,  158,  203,  301,  326 

Product 8 

Progressions,  arithmetic .     .     .  334 
"  geometric  .     .     .  340 

Proportion 169 

Proportional,  directly,  inversely,  169 

Quadratic  equations    ....  289 

Radical  equations 261 

Radicals 242 

Ratio 167 

Rational 96,  242,  376 

Rationalizing  denominators  .     .  252 

Real  numbers 218 

Reciprocal 139 

Reciprocal  equations  ....  497 
Relation  between  roots  and  co- 

elficients 340 

Remainder  theorem     .     .     267,  435 
Removal  of  parenthesis    ...     30 

Root 11 

Roots  of  an  equation  ....     65 
"      character  of .     .     .     .     .  327 

"      location  of 456 

"      multiple 437 

"      number  of 437 

"  sum  and  product  of  .  .  329 
Rows  of  a  determinant  .  .  .  402 
Rule  of  signs 37,  53 

Secondary  diagonal    ....  402 
Series 334 


662 


INDEX 


Series,  arithmetic 548 

"       fractional    .."...  558 

"       summation  of  ...     .  642 

Signs  of  aggregation    .     .     .     .     12 

Similar  and  dissimilar  terms     .     20 

Similar  surds 242 

Simple  equations 05 

Simultaneous  equations  .  184,  424 
Square  of  binomial  ....  47 
"  of  polynomial ....  49 
Square  root  ....  220,  221,  258 
Standard  reciprocal  equation    .  498 

Substitution 185 

Subtraction 25 

"  graphic     ....  382 

Subtrahend 25 

Sum,  algebraic 20 

Summation  of  series    ....  542 

Superior  limit 456 

Surds 242 

Symmetrical  function ....  443 
Symmetric  equations  .  .  312,  319 
Synthetic  division  .     ...     .     .431 


Table  of  logarithms     ....  528 

Term 12 

"      absolute 312 

Theorem,  binomial      .     .     .     .  348 

Theory  of  equations    ....  431 

Third  proportional 169 

Transformation  of  equations     .  448 

Transposition 65 

Trinomial 13 

Unknown  numbers     ....  1 

Value,  absolute 5 

Vanishing  fractions     ....  367 

Variable 272,  364 

"        dependent     ....  364 

*'        independent  ....  364 

Variation  .  ) 452,  516 

"             inverse      ....  517 

joint 517 

Vinculum 12 

Zero  exponent 232 


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